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'' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #66 to #67: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='differential_geometry'>Differential geometry</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a> Introductions <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>from point-set topology to differentiable manifolds</a> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometry of physics</a>: <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+coordinate+systems'>coordinate systems</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+sets'>smooth spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+manifolds+and+orbifolds'>manifolds</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+homotopy+types'>smooth homotopy types</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+supergeometry'>supergeometry</a> Differentials <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/chain+rule'>chain rule</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable function</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>infinitesimal space</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinitesimally+thickened+point'>infinitesimally thickened point</a>, <a class='existingWikiWord' href='/nlab/show/diff/amazing+right+adjoint'>amazing right adjoint</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/V-manifold'>V-manifolds</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>, <a class='existingWikiWord' href='/nlab/show/diff/atlas'>atlas</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/analytic+manifold'>analytic manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+manifold'>formal smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+smooth+manifold'>derived smooth manifold</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth space</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/frame+bundle'>frame bundle</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/multivector+field'>multivector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a>; </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+in+synthetic+differential+geometry'>differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+complex'>Dolbeault complex</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/pullback+of+a+differential+form'>pullback of differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/invariant+differential+form'>invariant differential form</a>, <a class='existingWikiWord' href='/nlab/show/diff/Maurer-Cartan+form'>Maurer-Cartan form</a>, <a class='existingWikiWord' href='/nlab/show/diff/horizontal+differential+form'>horizontal differential form</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cogerm+differential+form'>cogerm differential form</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/integration+of+differential+forms'>integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/local+diffeomorphism'>local diffeomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/submersion'>submersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+morphism'>formally smooth morphism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/immersion'>immersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+unramified+morphism'>formally unramified morphism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a>, <a class='existingWikiWord' href='/nlab/show/diff/crystal'>crystal</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+disk+bundle'>infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras'>embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+Serre-Swan+theorem'>smooth Serre-Swan theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hadamard+lemma'>Hadamard lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Borel%27s+theorem'>Borel's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Boman%27s+theorem'>Boman's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitney+extension+theorem'>Whitney extension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Steenrod+approximation+theorem'>Steenrod-Wockel approximation theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Kostant-Rosenberg+theorem'>Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kock-Lawvere+axiom'>Kock-Lawvere axiom</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+smooth+topos'>super smooth topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/microlinear+space'>microlinear space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/integration+axiom'>integration axiom</a> </li> </ul> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a> <ul> <li> (<a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/sharp+modality'>sharp modality</a>) <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='0em' rspace='thinmathspace'>esh</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+object'>codiscrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/concrete+object'>concrete object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/points-to-pieces+transform'>points-to-pieces transform</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos+--+structures'>structures in cohesion</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/dR-shape+modality'>dR-shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/dR-flat+modality'>dR-flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>esh</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding='application/x-tex'>\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+cohesive+%28infinity%2C1%29-topos'>infinitesimal cohesion</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+modality'>classical modality</a></li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/tangent+cohesive+%28%E2%88%9E%2C1%29-topos'>tangent cohesion</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology diagram</a></li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+%28infinity%2C1%29-topos'>differential cohesion</a> <ul> <li> (<a class='existingWikiWord' href='/nlab/show/diff/reduction+modality'>reduction modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+shape+modality'>infinitesimal shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+flat+modality'>infinitesimal flat modality</a>) <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/reduced+object'>reduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreduced+object'>coreduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+object'>formally smooth object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale map</a> </li> <li> <a href='cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion'>structures in differential cohesion</a> </li> </ul> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>graded differential cohesion</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>fermionic modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>bosonic modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/rheonomy+modality'>rheonomy modality</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/orbifold+cohomology'>singular cohesion</a> <div class='maruku-equation' id='Diagram'><math class='maruku-mathml' display='block' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant='normal'>R</mi><mspace width='negativethinmathspace' /><mspace width='negativethinmathspace' /><mi mathvariant='normal'>h</mi></mtd> <mtd><mover><mrow /><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow /><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>cohesive</mi></mover></mtd> <mtd><mo lspace='0em' rspace='thinmathspace'>esh</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow /><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div> Models <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Models+for+Smooth+Infinitesimal+Analysis'>Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/C%5E%E2%88%9E-ring'>smooth algebra</a> (<math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>C^\infty</annotation></semantics></math>-ring) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+locus'>smooth locus</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Fermat+theory'>Fermat theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>formal smooth ∞-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>super formal smooth ∞-groupoid</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>Lie n-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+2-group'>Lie 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-group</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/D-module'>D-module</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/jet+bundle'>jet bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/variational+bicomplex'>variational bicomplex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+complex'>Euler-Lagrange complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equation</a>, <a class='existingWikiWord' href='/nlab/show/diff/De+Donder-Weyl-Hamilton+equation'>de Donder-Weyl formalism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle'>connection on a bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+smooth+principal+infinity-bundle'>connection on an ∞-bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/ordinary+differential+cohomology'>ordinary differential cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/holonomy'>holonomy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+gauge+field'>higher gauge theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilson+surface'>Wilson surface</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a> (<a class='existingWikiWord' href='/nlab/show/diff/super-Cartan+geometry'>super</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher</a>) <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a>, (<a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+G-structure'>torsion of a G-structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+geometry'>Euclidean geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+geometry'>hyperbolic geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/elliptic+geometry'>elliptic geometry</a> </li> <li> (<a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+structure'>orthogonal structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+vector+field'>Killing vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+spinor'>Killing spinor</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super-spacetime</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a> </li> </ul> </div> <h4 id='higher_geometry'>Higher geometry</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>higher geometry</a> / <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a> Ingredients <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+topos+theory'>higher topos theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> </li> </ul> Concepts <ul> <li> geometric <a class='existingWikiWord' href='/nlab/show/diff/big+and+little+toposes'>little</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a>es <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>geometry (for structured (∞,1)-toposes)</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+scheme'>generalized scheme</a> </li> </ul> </li> <li> geometric <a class='existingWikiWord' href='/nlab/show/diff/big+and+little+toposes'>big</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a>es <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>function algebras on ∞-stacks</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+infinity-stack'>geometric ∞-stacks</a></li> </ul> </li> </ul> Constructions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/loop+space+object'>loop space object</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space+object'>free loop space object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>of a locally ∞-connected (∞,1)-topos</a> </li> </ul> Examples <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+algebraic+geometry'>derived algebraic geometry</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+%28infinity%2C1%29-site'>étale (∞,1)-site</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Hochschild cohomology</a> of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>s </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/dg-geometry'>dg-geometry</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/dg-scheme'>dg-scheme</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/schematic+homotopy+type'>schematic homotopy type</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+noncommutative+geometry'>derived noncommutative geometry</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/noncommutative+geometry'>noncommutative geometry</a></li> </ul> </li> <li> derived smooth geometry <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+smooth+manifold'>derived smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+manifold'>dg-manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid'>∞-Lie algebroid</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+symplectic+geometry'>higher symplectic geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher Klein geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher Cartan geometry</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Jones' theorem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Deligne-Kontsevich conjecture</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality+for+geometric+stacks'>Tannaka duality for geometric stacks</a> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#scope'>Scope</a></li><li><a href='#generalized_smooth_spaces_from_pov'>Generalized smooth spaces from <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>POV</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#References'>References</a><ul><li><a href='#ReferencesDiffGeometryOfCurvesAndSurfaces'>Diff geometry of curves and surfaces</a></li><li><a href='#general_but_traditional_diff_geometry'>General but traditional diff geometry</a></li><li><a href='#higher_diff_geometry'>Higher diff geometry</a></li><li><a href='#derived_diff_geometry'>Derived diff geometry</a></li></ul></li></ul></div> <h2 id='scope'>Scope</h2> Differential geometry is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces…) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifolds</a> modelled locally on <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector spaces</a>. Techniques of differential calculus can be further stretched to <a class='existingWikiWord' href='/nlab/show/diff/generalized+smooth+space'>generalized smooth spaces</a>. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, together with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>. <h2 id='generalized_smooth_spaces_from_pov'>Generalized smooth spaces from <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>POV</h2> See also <a class='existingWikiWord' href='/nlab/show/diff/generalized+smooth+space'>generalized smooth space</a>. Finite-dimensional differential geometry is the <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a> modeled on <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian spaces</a> and <a class='existingWikiWord' href='/nlab/show/diff/smooth+map'>smooth functions</a> between them. Formally, it is the <a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>geometry</a> modeled on the <a class='existingWikiWord' href='/nlab/show/diff/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>pre-geometry</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G} = </annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a>. This includes a sequence of concepts of <a class='existingWikiWord' href='/nlab/show/diff/generalized+smooth+space'>generalized smooth spaces</a>: <ul> <li> A <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> (see there for details) is a locally <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding='application/x-tex'>CartSp</annotation></semantics></math>-representable object in the <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sh</mi><mo stretchy='false'>(</mo><mi>CartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh(CartSp)</annotation></semantics></math>. </li> <li> A <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a> (see there) is a <a class='existingWikiWord' href='/nlab/show/diff/concrete+sheaf'>concrete sheaf</a> in the <a class='existingWikiWord' href='/nlab/show/diff/cohesive+topos'>cohesive topos</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sh</mi><mo stretchy='false'>(</mo><mi>CartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh(CartSp)</annotation></semantics></math>. </li> <li> a <a class='existingWikiWord' href='/nlab/show/diff/Lie+groupoid'>Lie groupoid</a> is a locally representable object in the <a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-sheaf'>(2,1)-sheaf</a> <a class='existingWikiWord' href='/nlab/show/diff/2-topos'>(2,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>CartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh_{(2,1)}(CartSp)</annotation></semantics></math>; </li> <li> an <a class='existingWikiWord' href='/nlab/show/diff/Lie+n-groupoid'>∞-Lie groupoid</a> is an object in the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-sheaf (∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>CartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh_{(\infty,1)}(CartSp)</annotation></semantics></math>. </li> </ul> Similarly, standard models of <a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic differential geometry</a> in <a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>higher geometry</a> are modeled on the <a class='existingWikiWord' href='/nlab/show/diff/geometry+%28for+structured+%28infinity%2C1%29-toposes%29'>pre-geometry</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G} = </annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/FormalCartSp'>ThCartSp</a>. To wit, the <a class='existingWikiWord' href='/nlab/show/diff/cohesive+topos'>cohesive topos</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sh</mi><mo stretchy='false'>(</mo><mi>ThCartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh(ThCartSp)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a> called the <a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a>: <ul> <li> an <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>infinitesimal space</a> is a certain object in <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ThCartSp</mi></mrow><annotation encoding='application/x-tex'>ThCartSp</annotation></semantics></math>; </li> <li> an <a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid'>∞-Lie algebroid</a> is a certain object in the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-category of (∞,1)-sheaves</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>ThCartSp</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sh_{(\infty,1)}(ThCartSp)</annotation></semantics></math>. </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/trigonometry'>trigonometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry+of+curves+and+surfaces'>differential geometry of curves and surfaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic differential geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+differential+geometry'>higher differential geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+differential+geometry'>derived differential geometry</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/motivation+for+higher+differential+geometry'>higher differential geometry applied to plain differential geometry</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+homotopy+type+theory'>differential cohesive homotopy type theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+analysis'>geometric analysis</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/prequantum+geometry'>prequantum geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+prequantum+geometry'>higher prequantum geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/arithmetic+differential+geometry'>arithmetic differential geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundles+in+physics'>fiber bundles in physics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tangent'>tangent</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent vector</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a> </li> <li> some modern subfields of differential geometry include: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/contact+manifold'>contact geometry</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a>, </li> <li> Finsler geometry<a href='/nlab/new/Finsler+geometry'>?</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symmetric+space'>symmetric spaces</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%A9chet+manifold'>Fréchet manifolds</a> </li> </ul> </li> </ul> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <table><thead><tr><th>local model</th><th>global geometry</th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Klein+2-geometry'>Klein 2-geometry</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>Cartan 2-geometry</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher Klein geometry</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher Cartan geometry</a></td></tr> </tbody></table> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometries of physics</a> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math>(<a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>higher</a>) <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/site'>site</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>∞-sheaf ∞-topos</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+geometry'>discrete geometry</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>Point</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/geometrically+discrete+infinity-groupoid'>Discrete∞Grpd</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>SmoothSet</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>Smooth∞Grpd</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>formal geometry</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/FormalCartSp'>FormalCartSp</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+set'>FormalSmoothSet</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>FormalSmooth∞Grpd</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/supergeometry'>supergeometry</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/super+Cartesian+space'>SuperFormalCartSp</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/super+smooth+set'>SuperFormalSmoothSet</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>SuperFormalSmooth∞Grpd</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <h2 id='References'>References</h2> <blockquote> See also references at <a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a>. </blockquote> <h3 id='ReferencesDiffGeometryOfCurvesAndSurfaces'>Diff geometry of curves and surfaces</h3> The study of differential geometry goes back to the special case of <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry+of+curves+and+surfaces'>differential geometry of curves and surfaces</a>: the study of <a class='existingWikiWord' href='/nlab/show/diff/curve'>curves</a> and <a class='existingWikiWord' href='/nlab/show/diff/surface'>surfaces</a> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+differentiable+manifolds'>embedded</a> into <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^3</annotation></semantics></math>: <ul> <li id='Gauss1827'> <a class='existingWikiWord' href='/nlab/show/diff/Carl+Gauss'>Carl Friedrich Gauss</a>, General Investigations of Curved Surfaces (1827) (<a href='http://www.gutenberg.org/ebooks/36856'>Gutenberg</a>) </li> <li> Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall (1976) [[pdf](http://www2.ing.unipi.it/griff/files/dC.pdf)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Heinrich+W.+Guggenheimer'>Heinrich W. Guggenheimer</a>, Differential Geometry, Dover (1977) [[isbn:9780486634333](https://store.doverpublications.com/products/9780486634333), <a href='https://archive.org/details/differentialgeom0000gugg/'>ark:/13960/t9t22sk9n</a>] </li> <li> Victor A. Toponogov, Differential Geometry of Curves and Surfaces — A Concise Guide, Springer (2006) [[doi:10.1007/b137116](https://doi.org/10.1007/b137116)] </li> <li> Kristopher Tapp, Differential Geometry of Curves and Surfaces, Springer (2016) <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.1007/978-3-319-39799-3'>doi:10.1007/978-3-319-39799-3</a>, <a href='https://link.springer.com/content/pdf/10.1007/978-3-319-39799-3.pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li> Anton Petrunin, Sergio Zamora Barrera, What is differential geometry: curves and surfaces <math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/2012.11814'>arXiv:2012.11814</a><math class='maruku-mathml' display='inline' id='mathml_15746dcea0ec407c57f81d0bdc1f8cb4ec5edf21_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> <h3 id='general_but_traditional_diff_geometry'>General but traditional diff geometry</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Shoshichi+Kobayashi'>Shoshichi Kobayashi</a>, <a class='existingWikiWord' href='/nlab/show/diff/Katsumi+Nomizu'>Katsumi Nomizu</a>, Foundations of differential geometry, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprint: Wiley Classics Library (1996) [[ISBN:978-0-470-55558-3](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+2+Volume+Set-p-9780470555583), <a href='https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry'>Wikipedia entry</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michael+Spivak'>Michael Spivak</a>, <a class='existingWikiWord' href='/nlab/show/diff/Calculus+on+Manifolds'>Calculus on Manifolds</a> (1971) [[pdf](http://www.strangebeautiful.com/other-texts/spivak-calc-manifolds.pdf)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Victor+Guillemin'>Victor Guillemin</a>, <a class='existingWikiWord' href='/nlab/show/diff/Shlomo+Sternberg'>Shlomo Sternberg</a>, Geometric asymptotics, Mathematical Surveys and Monographs 14, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Yvonne+Choquet-Bruhat'>Yvonne Choquet-Bruhat</a>, <a class='existingWikiWord' href='/nlab/show/diff/C%C3%A9cile+DeWitt-Morette'>Cécile DeWitt-Morette</a>, Analysis, manifolds and physics, North Holland (1982, 2001) [[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)] <blockquote> (with an eye towards <a class='existingWikiWord' href='/nlab/show/diff/mathematical+physics'>mathematical physics</a>) </blockquote> </li> <li> Thomas A. Ivey and J.M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics 61, AMS (2003) [[doi:10.1090/gsm/061](https://doi.org/10.1090/gsm/061), <a href='https://people.tamu.edu/~jml//EDSpublic.pdf'>pdf</a>, <a href='https://people.tamu.edu/~jml//EDSpublic.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michael+Spivak'>Michael Spivak</a>, A comprehensive introduction to differential geometry, 5 Volumes 2005 </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/M+M+Postnikov'>M M Postnikov</a>, Lectures on geometry 2001 (6 vols.: 1 “Analytic geometry”, 2 “Linear algebra”, 3 “Diff. manifolds”; 4 “Diff. geometry” (covers extensively fibre bundles and connections); 5 “Lie groups”; 6 “Riemannian geometry”) </li> <ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/Sigurdur+Helgason'>Sigurdur Helgason</a>, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [[ams:gsm-34](https://bookstore.ams.org/gsm-34)] </li></ins><ins class='diffins'> </ins><li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+Michor'>Peter W. Michor</a>, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008) [[pdf](https://www.mat.univie.ac.at/~michor/dgbook.pdf)] </li> <li id='Lee12'> <a class='existingWikiWord' href='/nlab/show/diff/John+Lee'>John Lee</a>, Introduction to Smooth Manifolds, Springer (2012) [[doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), <a href='https://sites.math.washington.edu/~lee/Books/ISM/'>book webpage</a>, <a href='https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Mikhail+O.+Katanaev'>Mikhail O. Katanaev</a>, Geometrical methods in mathematical physics (in Russian) [[arXiv1311.0733](https://arxiv.org/abs/1311.0733)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Loring+Tu'>Loring Tu</a>, Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer (2017) [[ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824)] </li> <li id='GallierQuaintance20a'> <a class='existingWikiWord' href='/nlab/show/diff/Jean+Gallier'>Jean Gallier</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jocelyn+Quaintance'>Jocelyn Quaintance</a>, Differential Geometry and Lie Groups – A computational perspective, Geometry and Computing 12, Springer (2020) [[doi:10.1007/978-3-030-46040-2](https://doi.org/10.1007/978-3-030-46040-2), <a href='https://www.cis.upenn.edu/~jean/gbooks/manif.html'>webpage</a>] </li> <li id='GallierQuaintance20b'> <a class='existingWikiWord' href='/nlab/show/diff/Jean+Gallier'>Jean Gallier</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jocelyn+Quaintance'>Jocelyn Quaintance</a>, Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), <a href='https://www.cis.upenn.edu/~jean/gbooks/manif.html'>webpage</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Andrzej+Derdzinski'>Andrzej Derdzinski</a>, Differential Geometry, lecture notes [[pdf](https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf), <a class='existingWikiWord' href='/nlab/files/Derdzinski-DifferentialGeometry.pdf' title='pdf'>pdf</a>] </li> </ul> With emphasis on <a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structures</a>: <ul> <li id='Sternberg64'><a class='existingWikiWord' href='/nlab/show/diff/Shlomo+Sternberg'>Shlomo Sternberg</a>, Lectures on differential geometry, Prentice-Hall (1964), AMS (1983) [ISBNJ:978-0-8218-1385-0, ams:chel-316, ark:/13960/t1pg9dv6k]</li> </ul> With emphasis on <a class='existingWikiWord' href='/nlab/show/diff/natural+bundle'>natural bundles</a>: <ul> <li id='KolarSlovakMichor93'><a class='existingWikiWord' href='/nlab/show/diff/Ivan+Kol%C3%A1%C5%99'>Ivan Kolář</a>, <a class='existingWikiWord' href='/nlab/show/diff/Peter+Michor'>Peter Michor</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jan+Slov%C3%A1k'>Jan Slovák</a>, <a class='existingWikiWord' href='/nlab/show/diff/Natural+operations+in+differential+geometry'>Natural operations in differential geometry</a>, Springer (1993) [[book webpage](http://www.emis.de/monographs/KSM/), <a href='https://link.springer.com/book/10.1007/978-3-662-02950-3'>doi:10.1007/978-3-662-02950-3</a> <a href='http://www.emis.de/monographs/KSM/kmsbookh.pdf'>pdf</a>]</li> </ul> With emphasis on <a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Richard+Sharpe'>Richard W. Sharpe</a>, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduare Texts in Mathematics 166, Springer (1997) [[ISBN:9780387947327](https://link.springer.com/book/9780387947327)]</li> </ul> Lecture notes: <ul> <li id='Conrad'> <a class='existingWikiWord' href='/nlab/show/diff/Brian+Conrad'>Brian Conrad</a>, Handouts on Differential Geometry (<a href='http://math.stanford.edu/~conrad/diffgeomPage/handouts.html'>web</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Liviu+Nicolaescu'>Liviu Nicolaescu</a>, Lectures on the Geometry of Manifolds, 2018 (<a href='https://www3.nd.edu/~lnicolae/Lectures.pdf'>pdf</a>) </li> </ul> Introductions with an eye towards applications in (<a class='existingWikiWord' href='/nlab/show/diff/mathematical+physics'>mathematical</a>)<a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a>, specifically to <a class='existingWikiWord' href='/nlab/show/diff/gravity'>gravity</a> and <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Theodore+Frankel'>Theodore Frankel</a>, <a class='existingWikiWord' href='/nlab/show/diff/The+Geometry+of+Physics+-+An+Introduction'>The Geometry of Physics - An Introduction</a>, Cambridge University Press (1997, 2004, 2012) [[doi:10.1017/CBO9781139061377](https://doi.org/10.1017/CBO9781139061377), <a href='http://www.math.ucsd.edu/~tfrankel/'>website</a> with errata and preface for 3rd edition)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Chris+Isham'>Chris Isham</a>: <a class='existingWikiWord' href='/nlab/show/diff/Modern+Differential+Geometry+for+Physicists'>Modern Differential Geometry for Physicists</a>, Lecture Notes in Physics 61, World Scientific (1999, 2001) [[doi:10.1142/0894](https://doi.org/10.1142/0894), <a href='https://cdn.preterhuman.net/texts/science_and_technology/physics/Mathematical_Physics/Modern%20Differential%20Geometry%20for%20Physicists%202nd%20ed.,%20-%20C.%20Isham.pdf'>pdf</a>, <a href='https://homepages.uc.edu/~herronda/Diff_Geometry/DG4physics.pdf'>pdf</a>] </li> </ul> A discussion in the context of <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher spaces</a> and <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological spaces</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Andreas+Kriegl'>Andreas Kriegl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Peter+Michor'>Peter Michor</a>, <a class='existingWikiWord' href='/nlab/show/diff/The+Convenient+Setting+of+Global+Analysis'>The convenient setting of global analysis</a>, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. 618 pages</li> </ul> See also <ul> <li> Sigmundur Gudmundsson, An Introduction to Riemannian Geometry (<a href='http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.pdf'>pdf</a>) </li> <li> Wikipedia, <a href='http://en.wikipedia.org/wiki/Differential_geometry'>differential geometry</a> </li> </ul> <h3 id='higher_diff_geometry'>Higher diff geometry</h3> See at <a class='existingWikiWord' href='/nlab/show/diff/higher+differential+geometry'>higher differential geometry</a>. <h3 id='derived_diff_geometry'>Derived diff geometry</h3> For <a class='existingWikiWord' href='/nlab/show/diff/derived+differential+geometry'>derived differential geometry</a> see <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Joyce'>Dominic Joyce</a>, D-manifolds and d-orbifolds: a theory of derived differential geometry (<a href='http://people.maths.ox.ac.uk/joyce/dmanifolds.html'>web</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <a class='existingWikiWord' href='/schreiber/show/diff/Seminar+on+derived+differential+geometry' title='schreiber'>Seminar on derived differential geometry</a> </li> </ul> </div> <div class="revisedby"> Last revised on July 11, 2024 at 10:45:43. 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