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Geodesic

From Wikipedia, the free encyclopedia

Straight path on a curved surface or a Riemannian manifold

This article is about geodesics in general. For geodesics in general relativity, seeGeodesic (general relativity). For the study of Earth's shape, seeGeodesy. For the application on Earth, seeEarth geodesic. For other uses, seeGeodesic (disambiguation).

Klein quartic with 28 geodesics(marked by 7 colors and 4 patterns)

Ingeometry, ageodesic (/ˌdʒiː.əˈdɛsɪk,-oʊ-,-ˈdiːsɪk,-zɪk/)^[1]^[2] is acurve representing in some sense the locally^[a] shortest^[b] path (arc) between two points in asurface, or more generally in aRiemannian manifold. The term also has meaning in anydifferentiable manifold with aconnection. It is a generalization of the notion of a "straight line".

The noungeodesic and the adjectivegeodetic come fromgeodesy, the science of measuring the size and shape ofEarth, though many of the underlying principles can be applied to anyellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth'ssurface. For aspherical Earth, it is asegment of agreat circle (see alsogreat-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, ingraph theory, one might consider ageodesic between twovertices/nodes of agraph.

In aRiemannian manifold or submanifold, geodesics are characterised by the property of having vanishinggeodesic curvature. More generally, in the presence of anaffine connection, a geodesic is defined to be a curve whosetangent vectors remain parallel if they aretransported along it. Applying this to theLevi-Civita connection of aRiemannian metric recovers the previous notion.

Geodesics are of particular importance ingeneral relativity. Timelikegeodesics in general relativity describe the motion offree falling test particles.

Introduction

A locally shortest path between two given points in a curved space, assumed^[b] to be aRiemannian manifold, can be defined by using theequation for thelength of acurve (a functionf from anopen interval ofR to the space), and then minimizing this length between the points using thecalculus of variations. This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance fromf(s) tof(t) along the curve equals |s−t|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization).^{[citation needed]} Intuitively, one can understand this second formulation by noting that anelastic band stretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.

It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic.

A contiguous segment of a geodesic is again a geodesic.

In general, geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are onlylocally the shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on agreat circle between two points on a sphere is a geodesic but not the shortest path between the points. The map $t\to t^{2}$ from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study ofRiemannian geometry and more generallymetric geometry. Ingeneral relativity, geodesics inspacetime describe the motion ofpoint particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbitingsatellite, or the shape of aplanetary orbit are all geodesics^[c] in curved spacetime. More generally, the topic ofsub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case ofRiemannian manifolds. The articleLevi-Civita connection discusses the more general case of apseudo-Riemannian manifold andgeodesic (general relativity) discusses the special case of general relativity in greater detail.

Examples

Ageodesic on a triaxial ellipsoid.

If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.

The most familiar examples are the straight lines inEuclidean geometry. On asphere, the images of geodesics are thegreat circles. The shortest path from pointA to pointB on a sphere is given by the shorterarc of the great circle passing throughA andB. IfA andB areantipodal points, then there areinfinitely many shortest paths between them.Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

Triangles

See also:Gauss–Bonnet theorem § For triangles, andToponogov's theorem

A geodesic triangle on the sphere.

Ageodesic triangle is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics aregreat circle arcs, forming aspherical triangle.

Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

Metric geometry

Inmetric geometry, a geodesic is a curve which is everywherelocally adistance minimizer. More precisely, acurveγ :I →M from an intervalI of the reals to themetric spaceM is ageodesic if there is aconstantv ≥ 0 such that for anyt ∈I there is a neighborhoodJ oft inI such that for anyt₁, t₂ ∈J we have

d(\gamma (t_{1}),\gamma (t_{2}))=v\left|t_{1}-t_{2}\right|.

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped withnatural parameterization, i.e. in the above identityv = 1 and

d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|.

If the last equality is satisfied for allt₁,t₂ ∈I, the geodesic is called aminimizing geodesic orshortest path.

In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in alength metric space are joined by a minimizing sequence ofrectifiable paths, although this minimizing sequence need not converge to a geodesic. Themetric Hopf-Rinow theorem provides situations where a length space is automatically a geodesic space.

Common examples of geodesic metric spaces that are often not manifolds includemetric graphs, (locally compact) metricpolyhedral complexes, infinite-dimensionalpre-Hilbert spaces, andreal trees.

Riemannian geometry

In aRiemannian manifold $M {\displaystyle M}$ withmetric tensor $g {\displaystyle g}$ , the length $L {\displaystyle L}$ of a continuously differentiable curve $\gamma :[a,b]\to M$ is defined by

L(\gamma )=\int _{a}^{b}{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt.

The distance $d(p,q)$ between two points $p {\displaystyle p}$ and $q {\displaystyle q}$ of $M {\displaystyle M}$ is defined as theinfimum of the length taken over all continuous, piecewise continuously differentiable curves $\gamma :[a,b]\to M$ such that $\gamma (a)=p$ and $\gamma (b)=q$ . In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics.

Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the followingaction orenergy functional

E(\gamma )={\frac {1}{2}}\int _{a}^{b}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt.

All minima of $E {\displaystyle E}$ are also minima of $L {\displaystyle L}$ , but $L {\displaystyle L}$ is a bigger set since paths that are minima of $L {\displaystyle L}$ can be arbitrarily re-parameterized (without changing their length), while minima of $E {\displaystyle E}$ cannot.For a piecewise $C^{1}$ curve (more generally, a $W^{1,2}$ curve), theCauchy–Schwarz inequality gives

L(\gamma )^{2}\leq 2(b-a)E(\gamma )

with equality if and only if $g(\gamma ',\gamma ')$ is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of $E(\gamma )$ also minimize $L(\gamma )$ , because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of $E {\displaystyle E}$ is a more robust variational problem. Indeed, $E(\gamma )$ is a "convex function" of $\gamma$ , so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of the functional $L(\gamma )$ are generally not very regular, because arbitrary reparameterizations are allowed.

TheEuler–Lagrange equations of motion for the functional $E {\displaystyle E}$ are then given in local coordinates by

{\frac {d^{2}x^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}=0,

where $\Gamma _{\mu \nu }^{\lambda }$ are theChristoffel symbols of the metric. This is thegeodesic equation, discussedbelow.

Calculus of variations

Techniques of the classicalcalculus of variations can be applied to examine the energy functional $E {\displaystyle E}$ . Thefirst variation of energy is defined in local coordinates by

\delta E(\gamma )(\varphi )=\left.{\frac {\partial }{\partial t}}\right|_{t=0}E(\gamma +t\varphi ).

Thecritical points of the first variation are precisely the geodesics. Thesecond variation is defined by

\delta ^{2}E(\gamma )(\varphi ,\psi )=\left.{\frac {\partial ^{2}}{\partial s\,\partial t}}\right|_{s=t=0}E(\gamma +t\varphi +s\psi ).

In an appropriate sense, zeros of the second variation along a geodesic $\gamma$ arise alongJacobi fields. Jacobi fields are thus regarded as variations through geodesics.

By applying variational techniques fromclassical mechanics, one can also regardgeodesics as Hamiltonian flows. They are solutions of the associatedHamilton equations, with(pseudo-)Riemannian metric taken asHamiltonian.

Affine geodesics

See also:Geodesics in general relativity

Ageodesic on asmooth manifold $M {\displaystyle M}$ with anaffine connection $\nabla$ is defined as acurve $\gamma (t)$ such thatparallel transport along the curve preserves the tangent vector to the curve, so

\nabla _{\dot {\gamma }}{\dot {\gamma }}=0

1

at each point along the curve, where ${\dot {\gamma }}$ is the derivative with respect to $t {\displaystyle t}$ . More precisely, in order to define the covariant derivative of ${\dot {\gamma }}$ it is necessary first to extend ${\dot {\gamma }}$ to a continuously differentiablevector field in anopen set. However, the resulting value of (1) is independent of the choice of extension.

Usinglocal coordinates on $M {\displaystyle M}$ , we can write thegeodesic equation (using thesummation convention) as

{\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ ,

where $\gamma ^{\mu }=x^{\mu }\circ \gamma (t)$ are the coordinates of the curve $\gamma (t)$ and $\Gamma _{\mu \nu }^{\lambda }$ are theChristoffel symbols of the connection $\nabla$ . This is anordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view ofclassical mechanics, geodesics can be thought of as trajectories offree particles in a manifold. Indeed, the equation $\nabla _{\dot {\gamma }}{\dot {\gamma }}=0$ means that theacceleration vector of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by gravity.

Existence and uniqueness

Thelocal existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with anaffine connection exist, and are unique. More precisely:

For any pointp inM and for any vectorV inT_pM (thetangent space toM atp) there exists a unique geodesic

\gamma \,

:I →M such that

\gamma (0)=p\,

and

{\dot {\gamma }}(0)=V,

whereI is a maximalopen interval inR containing 0.

The proof of this theorem follows from the theory ofordinary differential equations, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from thePicard–Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ dependssmoothly on bothp and V.

In general,I may not be all ofR as for example for an open disc inR². Anyγ extends to all ofℝ if and only ifM isgeodesically complete.

Geodesic flow

Geodesicflow is a localR-action on thetangent bundleTM of a manifoldM defined in the following way

G^{t}(V)={\dot {\gamma }}_{V}(t)

wheret ∈ R,V ∈ TM and $\gamma _{V}$ denotes the geodesic with initial data ${\dot {\gamma }}_{V}(0)=V$ . Thus, $G^{t}(V)=\exp(tV)$ is theexponential map of the vectortV. A closed orbit of the geodesic flow corresponds to aclosed geodesic on M.

On a (pseudo-)Riemannian manifold, the geodesic flow is identified with aHamiltonian flow on the cotangent bundle. TheHamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against thecanonical one-form. In particular the flow preserves the (pseudo-)Riemannian metric $g {\displaystyle g}$ , i.e.

g(G^{t}(V),G^{t}(V))=g(V,V).\,

In particular, whenV is a unit vector, $\gamma _{V}$ remains unit speed throughout, so the geodesic flow is tangent to theunit tangent bundle.Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

Geodesic spray

Further information:Spray (mathematics) § Geodesic

The geodesic flow defines a family of curves in thetangent bundle. The derivatives of these curves define avector field on thetotal space of the tangent bundle, known as thegeodesicspray.

More precisely, an affine connection gives rise to a splitting of thedouble tangent bundle TTM intohorizontal andvertical bundles:

TTM=H\oplus V.

The geodesic spray is the unique horizontal vector fieldW satisfying

\pi _{*}W_{v}=v\,

at each pointv ∈ TM; hereπ_∗ : TTM → TM denotes thepushforward (differential) along the projectionπ : TM → M associated to the tangent bundle.

More generally, the same construction allows one to construct a vector field for anyEhresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf.Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy

H_{\lambda X}=d(S_{\lambda })_{X}H_{X}\,

for everyX ∈ TM \ {0} and λ > 0. Hered(S_λ) is thepushforward along the scalar homothety $S_{\lambda }:X\mapsto \lambda X.$ A particular case of a non-linear connection arising in this manner is that associated to aFinsler manifold.

Affine and projective geodesics

Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form

t\mapsto at+b

wherea andb are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of (1) are called geodesics withaffine parameter.

An affine connection isdetermined by its family of affinely parameterized geodesics, up totorsion (Spivak 1999, Chapter 6, Addendum I). The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if $\nabla ,{\bar {\nabla }}$ are two connections such that the difference tensor

D(X,Y)=\nabla _{X}Y-{\bar {\nabla }}_{X}Y

isskew-symmetric, then $\nabla$ and ${\bar {\nabla }}$ have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as $\nabla$ , but with vanishing torsion.

Geodesics without a particular parameterization are described by aprojective connection.

Computational methods

Efficient solvers for the minimal geodesic problem on surfaces have been proposed by Mitchell,^[3] Kimmel,^[4] Crane,^[5] and others.

Ribbon test

A ribbon "test" is a way of finding a geodesic on a physical surface.^[6] The idea is to fit a bit of paper around a straight line (a ribbon) onto a curved surface as closely as possible without stretching or squishing the ribbon (without changing its internal geometry).

For example, when a ribbon is wound as a ring around a cone, the ribbon would not lie on the cone's surface but stick out, so that circle is not a geodesic on the cone. If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.

Mathematically the ribbon test can be formulated as finding a mapping $f:N(\ell )\to S$ of aneighborhood $N {\displaystyle N}$ of a line $\ell$ in a plane into a surface $S {\displaystyle S}$ so that the mapping $f {\displaystyle f}$ "doesn't change the distances around $\ell$ by much"; that is, at the distance $\varepsilon$ from $l {\displaystyle l}$ we have $g_{N}-f^{*}(g_{S})=O(\varepsilon ^{2})$ where $g_{N}$ and $g_{S}$ aremetrics on $N {\displaystyle N}$ and $S {\displaystyle S}$ .

Examples of applications

This sectionneeds expansion. You can help byadding to it.(June 2014)

While geometric in nature, the idea of a shortest path is so general that it easily finds extensive use in nearly all sciences, and in some other disciplines as well.

Topology and geometric group theory

In a surface with negativeEuler characteristic, any (free) homotopy class determines a unique (closed) geodesic for ahyperbolic metric. These geodesics contribute significantly to the geometric understanding of the action ofmapping classes.
Geodesic metric spaces andlength spaces behave particularly well with isometricgroup actions (Švarc-Milnor lemma,Hopf-Rinow theorem,Morse lemma...). They are often an adequate framework for generalizing results from Riemannian geometry to constructions that reflect the geometry of a group. For instance,Gromov-hyperbolicity can be understood in terms of geodesic triangle thinness, andCAT(0) can be stated in terms of angles between geodesics.

Probability, statistics and machine learning

Optimal transport can be understood as the problem of finding geodesic paths in spaces of measures.
Ininformation geometry,divergences such as theKullback-Leibler divergence play a role analogous to that of a Riemannian metric, allowing analogies forconnections and geodesics.

Physics

Inclassical mechanics,trajectories minimize an energy according to theHamilton-Jacobi equation, which can re regarded as a similar idea to geodesics. In some special cases,the two notions actually coincide.
Relativity theory modelsspacetime as aLorentzian manifold, where light follows Lorentzian geodesics.

Biology

The study of how thenervous system optimizes muscular movement may be approached by endowing aconfiguration space of the body with aRiemannian metric that measures the effort, so that the problem can be stated in terms of geodesy.^[7]
Geodesic distance is often used to measure the length of paths for signal propagation in neurons.^[8]
The structures of geodesics in large molecules plays a role in the study ofprotein folds.^[9]

Engineering

Geodesics serve as the basis to calculate:

geodesic airframes; seegeodesic airframe orgeodetic airframe
geodesic structures – for examplegeodesic domes
horizontal distances on or near Earth; seeEarth geodesics
mapping images on surfaces, for rendering; seeUV mapping
robotmotion planning (e.g., when painting car parts); seeShortest path problem
geodesic shortest path (GSP) correction overPoisson surface reconstruction (e.g. indigital dentistry); without GSP reconstruction often results in self-intersections within the surface

See also

Introduction to the mathematics of general relativity
Clairaut's relation – Formula in classical differential geometryPages displaying short descriptions of redirect targets
Differentiable curve – Study of curves from a differential point of view
Differential geometry of surfaces
Geodesic circle
Hopf–Rinow theorem – Gives equivalent statements about the geodesic completeness of Riemannian manifolds
Intrinsic metric – Concept in geometry/topology
Isotropic line – Line along which a quadratic form applied to any two points' displacement is zero
Jacobi field
Morse theory – Analyzes the topology of a manifold by studying differentiable functions on that manifold
Zoll surface – Surface homeomorphic to a sphere
The spider and the fly problem – Recreational geodesics problem

Notes

^For two points on a sphere that are not antipodes, there are two great circle arcs of different lengths connecting them, both of which are geodesics.
^^a ^bFor apseudo-Riemannian manifold, e.g., aLorentzian manifold, the definition is more complicated.
^The path is a local maximum of the intervalk rather than a local minimum.

References

^"geodesic".Lexico UK English Dictionary.Oxford University Press. Archived fromthe original on 2020-03-16.
^"geodesic".Merriam-Webster.com Dictionary. Merriam-Webster.
^Mitchell, J.; Mount, D.; Papadimitriou, C. (1987)."The Discrete Geodesic Problem".SIAM Journal on Computing.16 (4):647–668.doi:10.1137/0216045.
^Kimmel, R.; Sethian, J. A. (1998)."Computing Geodesic Paths on Manifolds"(PDF).Proceedings of the National Academy of Sciences.95 (15):8431–8435.Bibcode:1998PNAS...95.8431K.doi:10.1073/pnas.95.15.8431.PMC 21092.PMID 9671694.Archived(PDF) from the original on 2022-10-09.
^Crane, K.; Weischedel, C.; Wardetzky, M. (2017)."The Heat Method for Distance Computation".Communications of the ACM.60 (11):90–99.doi:10.1145/3131280.S2CID 7078650.
^Michael Stevens (Nov 2, 2017),[1].
^Neilson, Peter D.; Neilson, Megan D.; Bye, Robin T. (2015-12-01)."A Riemannian geometry theory of human movement: The geodesic synergy hypothesis".Human Movement Science.44:42–72.doi:10.1016/j.humov.2015.08.010.ISSN 0167-9457.PMID 26302481.
^Beshkov, Kosio; Tiesinga, Paul (2022-02-01)."Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex".Biological Cybernetics.116 (1):53–68.doi:10.1007/s00422-021-00906-5.ISSN 1432-0770.PMID 34816322.
^Zanotti, Giuseppe; Guerra, Concettina (2003-01-16)."Is tensegrity a unifying concept of protein folds?".FEBS Letters.534 (1):7–10.Bibcode:2003FEBSL.534....7Z.doi:10.1016/S0014-5793(02)03853-X.ISSN 0014-5793.PMID 12527354.

Spivak, Michael (1999),A Comprehensive introduction to differential geometry (Volume 2), Houston, TX: Publish or Perish,ISBN 978-0-914098-71-3

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Further reading

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(July 2014) (Learn how and when to remove this message)

Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975),Introduction to General Relativity (2nd ed.), New York:McGraw-Hill,ISBN 978-0-07-000423-8.See chapter 2.
Abraham, Ralph H.;Marsden, Jerrold E. (1978),Foundations of mechanics, London: Benjamin-Cummings,ISBN 978-0-8053-0102-1.See section 2.7.
Jost, Jürgen (2002),Riemannian Geometry and Geometric Analysis, Berlin, New York:Springer-Verlag,ISBN 978-3-540-42627-1.See section 1.4.
Kobayashi, Shoshichi; Nomizu, Katsumi (1996),Foundations of Differential Geometry, vol. 1 (New ed.), Wiley-Interscience,ISBN 0-471-15733-3.
Landau, L. D.;Lifshitz, E. M. (1975),Classical Theory of Fields, Oxford: Pergamon,ISBN 978-0-08-018176-9.See section 87.
Misner, Charles W.;Thorne, Kip;Wheeler, John Archibald (1973),Gravitation, W. H. Freeman,ISBN 978-0-7167-0344-0
Ortín, Tomás (2004),Gravity and strings,Cambridge University Press,ISBN 978-0-521-82475-0. Note especially pages 7 and 10.
Volkov, Yu.A. (2001) [1994],"Geodesic line",Encyclopedia of Mathematics,EMS Press.
Weinberg, Steven (1972),Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, New York:John Wiley & Sons,ISBN 978-0-471-92567-5.See chapter 3.

External links

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Geodesics Revisited — Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on atorus), mechanics (brachistochrone) and optics (light beam in inhomogeneous medium).
Totally geodesic submanifold at the Manifold Atlas

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem

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