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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #9 to #10: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='riemannian_geometry'>Riemannian geometry</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a></strong></p> <h2 id='basic_definitions'>Basic definitions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Riemannian+manifold'>Riemannian manifold</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric</a>, <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/isometry+group'>isometry group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+Riemannian+metrics'>moduli space of Riemannian metrics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+manifold'>pseudo-Riemannian manifold</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/smooth+Lorentzian+space'>Lorentzian manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic'>geodesic</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic+convexity'>geodesic convexity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic+flow'>geodesic flow</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Levi-Civita+connection'>Levi-Civita connection</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Riemann+curvature'>Riemann curvature</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+metric+connection'>torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id='further_concepts'>Further concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+star+operator'>Hodge inner product</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hodge+star+operator'>Hodge star operator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gradient'>gradient</a>, <a class='existingWikiWord' href='/nlab/show/diff/gradient+flow'>gradient flow</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+conjecture'>Poincaré conjecture</a>-theorem</li> </ul> <h2 id='applications'>Applications</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gravity'>gravity</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a>, <a class='existingWikiWord' href='/nlab/show/diff/Einstein+equation'>Einstein equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a></li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/Riemannian+geometry+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='differential_geometry'>Differential geometry</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>from point-set topology to differentiable manifolds</a></p> <p><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></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/chain+rule'>chain rule</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable function</a></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/V-manifold'>V-manifolds</a></strong></p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/analytic+manifold'>analytic manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth space</a></strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/frame+bundle'>frame bundle</a></p> <ul> <li> <p><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>;</p> </li> <li> <p><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></p> <ul> <li> <p><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>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cogerm+differential+form'>cogerm differential form</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+of+differential+forms'>integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/submersion'>submersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+morphism'>formally smooth morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/immersion'>immersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+unramified+morphism'>formally unramified morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a>, <a class='existingWikiWord' href='/nlab/show/diff/crystal'>crystal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+disk+bundle'>infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+Serre-Swan+theorem'>smooth Serre-Swan theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hadamard+lemma'>Hadamard lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Borel%27s+theorem'>Borel's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boman%27s+theorem'>Boman's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+extension+theorem'>Whitney extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Steenrod+approximation+theorem'>Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Kostant-Rosenberg+theorem'>Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kock-Lawvere+axiom'>Kock-Lawvere axiom</a></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microlinear+space'>microlinear space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+axiom'>integration axiom</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/points-to-pieces+transform'>points-to-pieces transform</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos+--+structures'>structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dR-shape+modality'>dR-shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> <p><math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+cohesive+%28infinity%2C1%29-topos'>infinitesimal cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+modality'>classical modality</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/tangent+cohesive+%28%E2%88%9E%2C1%29-topos'>tangent cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology diagram</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+%28infinity%2C1%29-topos'>differential cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/reduction+modality'>reduction modality</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale map</a></p> </li> <li> <p><a href='cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion'>structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>graded differential cohesion</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>fermionic modality</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> <p><math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/orbifold+cohomology'>singular cohesion</a></strong></p> <div class='maruku-equation' id='Diagram'><math class='maruku-mathml' display='block' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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> <p id='models_2'><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Models+for+Smooth+Infinitesimal+Analysis'>Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C%5E%E2%88%9E-ring'>smooth algebra</a> (<math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_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)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+locus'>smooth locus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fermat+theory'>Fermat theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>formal smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><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></strong></p> <ul> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/D-module'>D-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/jet+bundle'>jet bundle</a></p> </li> <li> <p><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></p> </li> <li> <p><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>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a></p> </li> </ul> <p><strong><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></strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holonomy'>holonomy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilson+surface'>Wilson surface</a></p> </li> </ul> <p><strong><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>)</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a>, (<a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher</a>)</p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+structure'>orthogonal structure</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super-spacetime</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#minimizing_geodesics'>Minimizing geodesics</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>On a <a class='existingWikiWord' href='/nlab/show/diff/Riemannian+manifold'>Riemannian manifold</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,g)</annotation></semantics></math>, a <strong>geodesic</strong> (or geodesic line, geodesic path) is a path <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x : I \to X</annotation></semantics></math>, for some (possibly infinite) <a class='existingWikiWord' href='/nlab/show/diff/interval'>interval</a> <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>, which locally minimizes distance.</p> <h2 id='definition'>Definition</h2> <p>One way to define a geodesic is as a <a class='existingWikiWord' href='/nlab/show/diff/critical+point'>critical point</a> of the <a class='existingWikiWord' href='/nlab/show/diff/functional'>functional</a> of length</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>↦</mo><msub><mo>∫</mo> <mi>I</mi></msub><msqrt><mrow><mi>g</mi><mo stretchy='false'>(</mo><mover><mi>x</mi><mo>⋅</mo></mover><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo>,</mo><mover><mi>x</mi><mo>⋅</mo></mover><mo stretchy='false'>(</mo><mi>t</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow></msqrt><mi>dt</mi></mrow><annotation encoding='application/x-tex'> (x : I \to M) \mapsto \int_I \sqrt{g(\stackrel{\cdot}{x}(t),\stackrel{\cdot}{x}(t))} dt </annotation></semantics></math></div> <p>on the appropriate space of curves <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>I \to X</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric</a> tensor. This implies that it satisfies the corresponding <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equation</a>s, which in this case means that the <a class='existingWikiWord' href='/nlab/show/diff/covariant+derivative'>covariant derivative</a> (for the <a class='existingWikiWord' href='/nlab/show/diff/Levi-Civita+connection'>Levi-Civita connection</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∇</mo> <mover><mi>x</mi><mo>⋅</mo></mover></msub><mover><mi>x</mi><mo>⋅</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'> \nabla_{\stackrel{\cdot}{x}} \stackrel{\cdot}{x} = 0 </annotation></semantics></math></div> <p>vanishes.</p> <p>In local coordinates, with <a class='existingWikiWord' href='/nlab/show/diff/Christoffel+symbols'>Christoffel symbol</a>s <math class='maruku-mathml' display='inline' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Γ</mi> <mi>jk</mi> <mi>i</mi></msubsup></mrow><annotation encoding='application/x-tex'>\Gamma^i_{jk}</annotation></semantics></math> the Euler-Lagrange equations for geodesics form a system</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_4e48b7974a37f4334c8c4420c8c019e39e96e17e_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><msup><mi>x</mi> <mi>i</mi></msup></mrow><mrow><msup><mi>dt</mi> <mn>2</mn></msup></mrow></mfrac><mo>+</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>jk</mi></munder><msubsup><mi>Γ</mi> <mi>jk</mi> <mi>i</mi></msubsup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mfrac><mrow><mi>d</mi><msup><mi>x</mi> <mi>j</mi></msup></mrow><mi>dt</mi></mfrac><mfrac><mrow><mi>d</mi><msup><mi>x</mi> <mi>k</mi></msup></mrow><mi>dt</mi></mfrac><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding='application/x-tex'> \frac{d^2 x^i}{dt^2} + \sum_{jk} \Gamma^i_{jk}(x) \frac{d x^j}{dt} \frac{d x^k}{dt} = 0. </annotation></semantics></math></div> <p>So this means that a curve is a <em>geodesic</em> if at every point its <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent vector</a> is the <a class='existingWikiWord' href='/nlab/show/diff/parallel+transport'>parallel transport</a> of the tangent vector at the start point along the curve.</p> <h2 id='minimizing_geodesics'>Minimizing geodesics</h2> <p>A geodesic may not <em>globally</em> minimize the distance between its end points. For instance, on a 2-dimensional <a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, geodesics are <a class='existingWikiWord' href='/nlab/show/diff/edge'>arcs</a> of <span class='newWikiWord'>great circle<a href='/nlab/new/great+circle'>?</a></span>s. Any two distinct non-antipodal points are connected by exactly two such geodesics, one shorter than the other (you can go from Los Angeles to Boston directly across North America, or the long way around the world).</p> <p>A geodesic which does globally minimize distance between its end points is called a <strong>minimizing geodesic</strong>. The length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold which makes it into a <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/prime+geodesic'>prime geodesic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic+completeness'>geodesic completeness</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/totally+geodesic+submanifold'>totally geodesic submanifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li><span><del class='diffdel'> en.wikipedia:</del></span><del class='diffmod'><a href='http://en.wikipedia.org/wiki/Geodesic'>geodesic</a></del><ins class='diffmod'><p>Wikipedia: <em><a href='http://en.wikipedia.org/wiki/Geodesic'>Geodesic</a></em></p></ins><ins class='diffins'> </ins></li> <li><span><del class='diffdel'> Springer</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Encyclopaedia+of+Mathematics'>eom</a></del><ins class='diffmod'><p>Springer <a class='existingWikiWord' href='/nlab/show/diff/Encyclopaedia+of+Mathematics'>eom</a>: Yu. A. Volkov, <a href='http://eom.springer.de/G/g044120.htm'>Geodesic line</a>; Yu. A. Volkov, <a href='http://eom.springer.de/G/g044060.htm'>Geodesic coordinates</a>; <a href='http://eom.springer.de/G/g044080.htm'>Geodesic distance</a>; V.A. Zalgaller, <a href='http://eom.springer.de/G/g044100.htm'>Geodesic geometry</a>; D.V. Anosov, <a href='http://eom.springer.de/G/g044090.htm'>Geodesic flow</a></p></ins><span><del class='diffdel'> :</del><del class='diffdel'> Yu.</del><del class='diffdel'> A.</del><del class='diffdel'> Volkov,</del></span><del class='diffdel'><a href='http://eom.springer.de/G/g044120.htm'>Geodesic line</a></del><del class='diffdel'>; Yu. A. Volkov, </del><del class='diffdel'><a href='http://eom.springer.de/G/g044060.htm'>Geodesic coordinates</a></del><del class='diffdel'>; </del><del class='diffdel'><a href='http://eom.springer.de/G/g044080.htm'>Geodesic distance</a></del><del class='diffdel'>; V.A. Zalgaller, </del><del class='diffdel'><a href='http://eom.springer.de/G/g044100.htm'>Geodesic geometry</a></del><del class='diffdel'>; D.V. Anosov, </del><del class='diffdel'><a href='http://eom.springer.de/G/g044090.htm'>Geodesic flow</a></del></li> <li><span><del class='diffdel'> Sh.</del><del class='diffdel'> Kobayashi,</del><del class='diffdel'> K.</del><del class='diffdel'> Nomidzu,</del></span><del class='diffmod'><em>Foundations of differential geometry</em></del><ins class='diffmod'><p>Sh. Kobayashi, K. Nomidzu, <em>Foundations of differential geometry</em>, vol 1, 1963, vol 2, 1969, Wiley Interscience; reedition 1996 in series Wiley Classics; Russian ed.: Nauka, Moscow 1981.</p></ins><span><del class='diffdel'> ,</del><del class='diffdel'> vol</del><del class='diffdel'> 1,</del><del class='diffdel'> 1963,</del><del class='diffdel'> vol</del><del class='diffdel'> 2,</del><del class='diffdel'> 1969,</del><del class='diffdel'> Wiley</del><del class='diffdel'> Interscience;</del><del class='diffdel'> reedition</del><del class='diffdel'> 1996</del><del class='diffdel'> in</del><del class='diffdel'> series</del><del class='diffdel'> Wiley</del><del class='diffdel'> Classics;</del><del class='diffdel'> Russian</del><del class='diffdel'> ed.:</del><del class='diffdel'> Nauka,</del><del class='diffdel'> Moscow</del><del class='diffdel'> 1981.</del></span></li> <li><span><del class='diffdel'> Arthur</del><del class='diffdel'> L.</del><del class='diffdel'> Besse,</del></span><del class='diffmod'><em>Einstein manifolds</em></del><ins class='diffmod'><p>Arthur L. Besse, <em>Einstein manifolds</em>, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag 1987, xii + 510 pp. (for a review see <a href='http://projecteuclid.org/euclid.bams/1183554925'>Bull. AMS</a> and <a>MR88f:53087</a>); reprinted 2008, Springer Classics in Math.</p></ins><span><del class='diffdel'> ,</del><del class='diffdel'> Ergebnisse</del><del class='diffdel'> der</del><del class='diffdel'> Mathematik</del><del class='diffdel'> und</del><del class='diffdel'> ihrer</del><del class='diffdel'> Grenzgebiete,</del><del class='diffdel'> Band</del><del class='diffdel'> 10,</del><del class='diffdel'> Springer-Verlag</del><del class='diffdel'> 1987,</del><del class='diffdel'> xii</del><del class='diffdel'> +</del><del class='diffdel'> 510</del><del class='diffdel'> pp.</del><del class='diffdel'> (for</del><del class='diffdel'> a</del><del class='diffdel'> review</del><del class='diffdel'> see</del></span><del class='diffdel'><a href='http://projecteuclid.org/euclid.bams/1183554925'>Bull. AMS</a></del><del class='diffdel'> and </del><del class='diffdel'><a>MR88f:53087</a></del><del class='diffdel'>); reprinted 2008, Springer Classics in Math.</del></li> </ul><ins class='diffins'> </ins><ins class='diffins'><p>On <a class='existingWikiWord' href='/nlab/show/diff/Maslov+index'>Maslov indices</a> and stability of geodesics:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Paolo+Piccione'>Paolo Piccione</a>, <a class='existingWikiWord' href='/nlab/show/diff/Alessandro+Portaluri'>Alessandro Portaluri</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+V.+Tausk'>Daniel V. Tausk</a>: <em>Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics</em>, Annals of Global Analysis and Geometry <strong>25</strong> (2004) 121–149 [<a href='https://doi.org/10.1023/B:AGAG.0000018558.65790.db'>doi:10.1023/B:AGAG.0000018558.65790.db</a>]</p> </li> <li id='PortaluriWuYang21'> <p><a class='existingWikiWord' href='/nlab/show/diff/Alessandro+Portaluri'>Alessandro Portaluri</a>, Li Wu, Ran Yang: <em>Linear instability for periodic orbits of non-autonomous Lagrangian systems</em>, Nonlinearity <strong>34</strong> 1 (2021) 237 [[arXiv:1907.05864](https://arxiv.org/abs/1907.05864)]</p> </li> </ul></ins> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 4, 2024 at 14:24:33. 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