Indifferential geometry, aRiemannian manifold is ageometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.Euclidean space, the-sphere,hyperbolic space, andsmooth surfaces in three-dimensional space, such asellipsoids andparaboloids, are all examples of Riemannianmanifolds. Riemannian manifolds are named after German mathematicianBernhard Riemann, who first conceptualized them.
Formally, aRiemannian metric (or just ametric) on asmooth manifold is a smoothly varying choice ofinner product for eachtangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques ofdifferential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature andparallel transport.
Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside theambient space. The same is true for anysubmanifold of Euclidean space of any dimension. AlthoughJohn Nash proved that every Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many instances, such as for hyperbolic space andprojective space, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics onLie groups andhomogeneous spaces are defined intrinsically by usinggroup actions to transport an inner product on a single tangent space to the entire manifold, and many special metrics such asconstant scalar curvature metrics andKähler–Einstein metrics are constructed intrinsically using tools frompartial differential equations.
Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of math, includinggeometric topology,complex geometry, andalgebraic geometry. Applications includephysics (especiallygeneral relativity andgauge theory),computer graphics,machine learning, andcartography. Generalizations of Riemannian manifolds includepseudo-Riemannian manifolds,Finsler manifolds, andsub-Riemannian manifolds.
In 1827,Carl Friedrich Gauss discovered that theGaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (thefirst fundamental form).[1] This result is known as theTheorema Egregium ("remarkable theorem" in Latin).
A map that preserves the local measurements of a surface is called alocal isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.
Riemannian manifolds and their curvature were first introduced non-rigorously byBernhard Riemann in 1854.[2] However, they would not be formalized until much later. In fact, the more primitive concept of asmooth manifold was first explicitly defined only in 1913 in a book byHermann Weyl.[2]
Élie Cartan introduced theCartan connection, one of the first concepts of aconnection.Levi-Civita defined theLevi-Civita connection, a special connection on a Riemannian manifold.
Albert Einstein used the theory ofpseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to developgeneral relativity. Specifically, theEinstein field equations are constraints on the curvature ofspacetime, which is a 4-dimensional pseudo-Riemannian manifold.
Let be asmooth manifold. For each point, there is an associated vector space called thetangent space of at. Vectors in are thought of as the vectors tangent to at.
However, does not come equipped with aninner product, a "measuring stick" that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts such a "measuring stick" on every tangent space.
ARiemannian metric on assigns to each apositive-definitesymmetric bilinear form (i.e. an inner product) in a smooth way (see the section on regularity below).[3] This induces a norm defined by. A smooth manifold endowed with a Riemannian metric is aRiemannian manifold, denoted.[3] A Riemannian metric is a special case of ametric tensor.
A Riemannian metric is not to be confused with the distance function of ametric space, which is also called a metric.
If are smoothlocal coordinates on, the vectors
form a basis of the vector space for any. Relative to this basis, one can define the Riemannian metric's components at each point by
These functions can be put together into an matrix-valued function on. The requirement that is a positive-definite inner product then says exactly that this matrix-valued function is asymmetricpositive-definite matrix at.
In terms of thetensor algebra, the Riemannian metric can be written in terms of thedual basis of the cotangent bundle as
The Riemannian metric iscontinuous if its components are continuous in any smooth coordinate chart The Riemannian metric issmooth if its components are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such asLipschitz Riemannian metrics ormeasurable Riemannian metrics.
There are situations ingeometric analysis in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). However, in this article, is assumed to be smooth unless stated otherwise.
In analogy to how an inner product on a vector space induces an isomorphism between a vector space and itsdual given by, a Riemannian metric induces an isomorphism of bundles between thetangent bundle and thecotangent bundle. Namely, if is a Riemannian metric, then
is a isomorphism ofsmooth vector bundles from the tangent bundle to the cotangent bundle.[5]
An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are calledisometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.
Specifically, if and are two Riemannian manifolds, adiffeomorphism is called anisometry if,[6] that is, if
for all and For example, translations and rotations are both isometries from Euclidean space (to be defined soon) to itself.
One says that a smooth map not assumed to be a diffeomorphism, is alocal isometry if every has an open neighborhood such that is an isometry (and thus a diffeomorphism).[6]
An oriented-dimensional Riemannian manifold has a unique-form called theRiemannian volume form.[7] The Riemannian volume form is preserved by orientation-preserving isometries.[8] The volume form gives rise to ameasure on which allows measurable functions to be integrated.[citation needed] If iscompact, thevolume of is.[7]
Let denote the standard coordinates on The (canonical)Euclidean metric is given by[9]
or equivalently
or equivalently by its coordinate functions
which together form the matrix
The Riemannian manifold is calledEuclidean space.
Let be a Riemannian manifold and let be animmersed submanifold or anembedded submanifold of. Thepullback of is a Riemannian metric on, and is said to be aRiemannian submanifold of.[10]
In the case where, the map is given by and the metric is just the restriction of to vectors tangent along. In general, the formula for is
where is thepushforward of by
Examples:
On the other hand, if already has a Riemannian metric, then the immersion (or embedding) is called anisometric immersion (orisometric embedding) if. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.[10]
Let and be two Riemannian manifolds, and consider theproduct manifold. The Riemannian metrics and naturally put a Riemannian metric on which can be described in a few ways.
For example, consider the-torus. If each copy of is given the round metric, the product Riemannian manifold is called theflat torus. As another example, the Riemannian product, where each copy of has the Euclidean metric, is isometric to with the Euclidean metric.
Let be Riemannian metrics on If are any positive smooth functions on, then is another Riemannian metric on
Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric.[13]
This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds areHausdorff andparacompact. The reason is that the proof makes use of apartition of unity.
Proof that every smooth manifold admits a Riemannian metric |
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Let be a smooth manifold and alocally finiteatlas so that are open subsets and are diffeomorphisms. Such an atlas exists because the manifold is paracompact. Let be a differentiablepartition of unity subordinate to the given atlas, i.e. such that for all. Define a Riemannian metric on by where Here is the Euclidean metric on and is itspullback along. While is only defined on, the product is defined and smooth on since. It takes the value 0 outside of. Because the atlas is locally finite, at every point the sum contains only finitely many nonzero terms, so the sum converges. It is straightforward to check that is a Riemannian metric. |
An alternative proof uses theWhitney embedding theorem to embed into Euclidean space and then pulls back the metric from Euclidean space to. On the other hand, theNash embedding theorem states that, given any smooth Riemannian manifold there is an embedding for some such that thepullback by of the standard Riemannian metric on is That is, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as theset of rotations of three-dimensional space and hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.
Anadmissible curve is a piecewise smooth curve whose velocity is nonzero everywhere it is defined. The nonnegative function is defined on the interval except for at finitely many points. The length of an admissible curve is defined as
The integrand is bounded and continuous except at finitely many points, so it is integrable. For a connected Riemannian manifold, define by
Theorem: is ametric space, and themetric topology on coincides with the topology on.[14]
Proof sketch that is a metric space, and the metric topology on agrees with the topology on |
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In verifying that satisfies all of theaxioms of a metric space, the most difficult part is checking that implies. Verification of the other metric space axioms is omitted. There must be some precompact open set aroundp which every curve fromp toq must escape. By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart aroundp, any curve fromp toq must first pass though a certain "inner radius." The assumed continuity of the Riemannian metricg only allows this "coordinate chart geometry" to distort the "true geometry" by some bounded factor. To be precise, let be a smooth coordinate chart with and Let be an open subset of with By continuity of and compactness of there is a positive number such that for any and any where denotes the Euclidean norm induced by the local coordinates. LetR denote.Now, given any admissible curve fromp toq, there must be some minimal such that clearly The length of is at least as large as the restriction of to So The integral which appears here represents the Euclidean length of a curve from 0 to, and so it is greater than or equal toR. So we conclude The observation about comparison between lengths measured byg and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of coincides with the original topological space structure of. |
Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function by any explicit means. In fact, if is compact, there always exist points where is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when is an ellipsoid.[citation needed]
If one works with Riemannian metrics that are merely continuous but possibly not smooth, the length of an admissible curve and the Riemannian distance function are defined exactly the same, and, as before, is ametric space and themetric topology on coincides with the topology on.[15]
Thediameter of the metric space is
TheHopf–Rinow theorem shows that if iscomplete and has finite diameter, it is compact. Conversely, if is compact, then the function has a maximum, since it is a continuous function on a compact metric space. This proves the following.
This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric. It is also not true thatany complete metric space of finite diameter must be compact; it matters that the metric space came from a Riemannian manifold.
An(affine) connection is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.
Let denote the space ofvector fields on. An(affine) connection
on is a bilinear map such that
The expression is called thecovariant derivative of with respect to.
Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called theLevi-Civita connection.
A connection is said topreserve the metric if
A connection istorsion-free if
where is theLie bracket.
ALevi-Civita connection is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.[17] Note that the definition of preserving the metric uses the regularity of.
If is a smooth curve, asmooth vector field along is a smooth map such that for all. The set of smooth vector fields along is a vector space under pointwise vector addition andscalar multiplication.[18] One can also pointwise multiply a smooth vector field along by a smooth function:
Let be a smooth vector field along. If is a smooth vector field on a neighborhood of the image of such that, then is called anextension of.
Given a fixed connection on and a smooth curve, there is a unique operator, called thecovariant derivative along, such that:[19]
Geodesics are curves with no intrinsic acceleration. Equivalently, geodesics are curves that locally take the shortest path between two points. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds. An ant living in a Riemannian manifold walking straight ahead without making any effort to accelerate or turn would trace out a geodesic.
Fix a connection on. Let be a smooth curve. Theacceleration of is the vector field along. If for all, is called ageodesic.[20]
For every and, there exists a geodesic defined on some open interval containing 0 such that and. Any two such geodesics agree on their common domain.[21] Taking the union over all open intervals containing 0 on which a geodesic satisfying and exists, one obtains a geodesic called amaximal geodesic of which every geodesic satisfying and is a restriction.[22]
Every curve that has the shortest length of any admissible curve with the same endpoints as is a geodesic (in a unit-speed reparameterization).[23]
The Riemannian manifold with its Levi-Civita connection isgeodesically complete if the domain of every maximal geodesic is.[25] The plane is geodesically complete. On the other hand, thepunctured plane with the restriction of the Riemannian metric from is not geodesically complete as the maximal geodesic with initial conditions, does not have domain.
TheHopf–Rinow theorem characterizes geodesically complete manifolds.
Theorem: Let be a connected Riemannian manifold. The following are equivalent:[26]
In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another.Parallel transport is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.[27]
Specifically, call a smooth vector field along a smooth curveparallel along if identically.[22] Fix a curve with and. to parallel transport a vector to a vector in along, first extend to a vector field parallel along, and then take the value of this vector field at.
The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on thepunctured plane. The curve the parallel transport is done along is the unit circle. Inpolar coordinates, the metric on the left is the standard Euclidean metric, while the metric on the right is. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.
Warning: This is parallel transport on the punctured planealong the unit circle, not parallel transporton the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.
The Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map.[28] The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.[29]
Fix a connection on. TheRiemann curvature tensor is the map defined by
where is theLie bracket of vector fields. The Riemann curvature tensor is a-tensor field.[30]
Fix a connection on. TheRicci curvature tensor is
where is the trace. The Ricci curvature tensor is a covariant 2-tensor field.[31]
The Ricci curvature tensor plays a defining role in the theory ofEinstein manifolds, which has applications to the study ofgravity. A (pseudo-)Riemannian metric is called anEinstein metric ifEinstein's equation
holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called anEinstein manifold.[32] Examples of Einstein manifolds include Euclidean space, the-sphere, hyperbolic space, andcomplex projective space with theFubini-Study metric.
A Riemannian manifold is said to haveconstant curvatureκ if everysectional curvature equals the numberκ. This is equivalent to the condition that, relative to any coordinate chart, theRiemann curvature tensor can be expressed in terms of themetric tensor as
This implies that theRicci curvature is given byRjk = (n − 1)κgjk and thescalar curvature isn(n − 1)κ, wheren is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is anEinstein manifold, thereby having constant scalar curvature. As found byBernhard Riemann in his 1854 lecture introducing Riemannian geometry, the locally defined Riemannian metric
has constant curvatureκ. Any two Riemannian manifolds of the same constant curvature arelocally isometric, and so it follows that any Riemannian manifold of constant curvatureκ can be covered by coordinate charts relative to which the metric has the above form.[33]
ARiemannian space form is a Riemannian manifold with constant curvature which is additionallyconnected andgeodesically complete. A Riemannian space form is said to be aspherical space form if the curvature is positive, aEuclidean space form if the curvature is zero, and ahyperbolic space form orhyperbolic manifold if the curvature is negative. In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature1,0, and−1 respectively. Furthermore, theKilling–Hopf theorem says that any simply connected spherical space form is homothetic to the sphere, any simply connected Euclidean space form is homothetic to Euclidean space, and any simply connected hyperbolic space form is homothetic to hyperbolic space.[33]
Using thecovering manifold construction, any Riemannian space form is isometric to thequotient manifold of a simply connected Riemannian space form, modulo a certain group action of isometries. For example, the isometry group of then-sphere is theorthogonal groupO(n + 1). Given any finitesubgroupG thereof in which only theidentity matrix possesses1 as aneigenvalue, the natural group action of the orthogonal group on then-sphere restricts to a group action ofG, with thequotient manifoldSn /G inheriting a geodesically complete Riemannian metric of constant curvature1. Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems ingroup theory. For instance, this can be used to show directly that every even-dimensional spherical space form is homothetic to the standard metric on either the sphere orreal projective space. There are many more odd-dimensional spherical space forms, although there are known algorithms for their classification. The list of three-dimensional spherical space forms is infinite but explicitly known, and includes thelens spaces and thePoincaré dodecahedral space.[34]
The case of Euclidean and hyperbolic space forms can likewise be reduced to group theory, based on study of the isometry group of Euclidean space and hyperbolic space. For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on theKlein bottle, theMöbius strip, thetorus, thecylinderS1 × ℝ, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic. The case of two-dimensional hyperbolic space forms is even more complicated, having to do withTeichmüller space. In three dimensions, the Euclidean space forms are known, while the geometry of hyperbolic space forms in three and higher dimensions remains an area of active research known ashyperbolic geometry.[35]
LetG be aLie group, such as thegroup of rotations in three-dimensional space. Using the group structure, any inner product on the tangent space at the identity (or any other particular tangent space) can be transported to all other tangent spaces to define a Riemannian metric. Formally, given an inner productge on the tangent space at the identity, the inner product on the tangent space at an arbitrary pointp is defined by
where for arbitraryx,Lx is the left multiplication mapG →G sending a pointy toxy. Riemannian metrics constructed this way areleft-invariant; right-invariant Riemannian metrics could be constructed likewise using the right multiplication map instead.
The Levi-Civita connection and curvature of a general left-invariant Riemannian metric can be computed explicitly in terms ofge, theadjoint representation ofG, and theLie algebra associated toG.[36] These formulas simplify considerably in the special case of a Riemannian metric which isbi-invariant (that is, simultaneously left- and right-invariant).[37] All left-invariant metrics have constant scalar curvature.
Left- and bi-invariant metrics on Lie groups are an important source of examples of Riemannian manifolds.Berger spheres, constructed as left-invariant metrics on thespecial unitary group SU(2), are among the simplest examples of thecollapsing phenomena, in which a simply connected Riemannian manifold can have small volume without having large curvature.[38] They also give an example of a Riemannian metric which has constant scalar curvature but which is notEinstein, or even of parallel Ricci curvature.[39] Hyperbolic space can be given a Lie group structure relative to which the metric is left-invariant.[40][41] Any bi-invariant Riemannian metric on a Lie group has nonnegative sectional curvature, giving a variety of such metrics: a Lie group can be given a bi-invariant Riemannian metric if and only if it is the product of acompact Lie group with anabelian Lie group.[42]
A Riemannian manifold(M,g) is said to behomogeneous if for every pair of pointsx andy inM, there is some isometryf of the Riemannian manifold sendingx toy. This can be rephrased in the language ofgroup actions as the requirement that the natural action of theisometry group is transitive. Every homogeneous Riemannian manifold is geodesically complete and has constant scalar curvature.[43]
Up to isometry, all homogeneous Riemannian manifolds arise by the following construction. Given a Lie groupG with compact subgroupK which does not contain any nontrivialnormal subgroup ofG, fix anycomplemented subspaceW of theLie algebra ofK within the Lie algebra ofG. If this subspace is invariant under the linear mapadG(k):W →W for any elementk ofK, thenG-invariant Riemannian metrics on thecoset spaceG/K are in one-to-one correspondence with those inner products onW which are invariant underadG(k):W →W for every elementk ofK.[44] Each such Riemannian metric is homogeneous, withG naturally viewed as a subgroup of the full isometry group.
The above example of Lie groups with left-invariant Riemannian metrics arises as a very special case of this construction, namely whenK is the trivial subgroup containing only the identity element. The calculations of the Levi-Civita connection and the curvature referenced there can be generalized to this context, where now the computations are formulated in terms of the inner product onW, the Lie algebra ofG, and the direct sum decomposition of the Lie algebra ofG into the Lie algebra ofK andW.[44] This reduces the study of the curvature of homogeneous Riemannian manifolds largely to algebraic problems. This reduction, together with the flexibility of the above construction, makes the class of homogeneous Riemannian manifolds very useful for constructing examples.
A connected Riemannian manifold(M,g) is said to besymmetric if for every pointp ofM there exists some isometry of the manifold withp as afixed point and for which the negation of thedifferential atp is theidentity map. Every Riemannian symmetric space is homogeneous, and consequently is geodesically complete and has constant scalar curvature. However, Riemannian symmetric spaces also have a much stronger curvature property not possessed by most homogeneous Riemannian manifolds, namely that theRiemann curvature tensor andRicci curvature areparallel. Riemannian manifolds with this curvature property, which could loosely be phrased as "constant Riemann curvature tensor" (not to be confused withconstant curvature), are said to belocally symmetric. This property nearly characterizes symmetric spaces;Élie Cartan proved in the 1920s that a locally symmetric Riemannian manifold which is geodesically complete andsimply-connected must in fact be symmetric.[45]
Many of the fundamental examples of Riemannian manifolds are symmetric. The most basic include the sphere andreal projective spaces with their standard metrics, along with hyperbolic space. The complex projective space,quaternionic projective space, andCayley plane are analogues of the real projective space which are also symmetric, as arecomplex hyperbolic space, quaternionic hyperbolic space, and Cayley hyperbolic space, which are instead analogues of hyperbolic space.Grassmannian manifolds also carry natural Riemannian metrics making them into symmetric spaces. Among the Lie groups with left-invariant Riemannian metrics, those which are bi-invariant are symmetric.[45]
Based on their algebraic formulation as special kinds of homogeneous spaces, Cartan achieved an explicit classification of symmetric spaces which areirreducible, referring to those which cannot be locally decomposed asproduct spaces. Every such space is an example of anEinstein manifold; among them only the one-dimensional manifolds have zero scalar curvature. These spaces are important from the perspective ofRiemannian holonomy. As found in the 1950s byMarcel Berger, any Riemannian manifold which is simply connected and irreducible is either a symmetric space or has Riemannian holonomy belonging to a list of only seven possibilities. Six of the seven exceptions to symmetric spaces in Berger's classification fall into the fields ofKähler geometry,quaternion-Kähler geometry,G2 geometry, andSpin(7) geometry, each of which study Riemannian manifolds equipped with certain extra structures and symmetries. The seventh exception is the study of 'generic' Riemannian manifolds with no particular symmetry, as reflected by the maximal possible holonomy group.[45]
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The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after atopological vector space; for example,Fréchet,Banach, andHilbert manifolds.
Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics:
Length of curves and the Riemannian distance function are defined in a way similar to the finite-dimensional case. The distance function, called thegeodesic distance, is always apseudometric (a metric that does not separate points), but it may not be a metric.[46] In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compactopen set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.
In the case of strong Riemannian metrics, one part of the finite-dimensional Hopf–Rinow still holds.
Theorem: Let be a strong Riemannian manifold. Then metric completeness (in the metric) implies geodesic completeness.[citation needed]
However, a geodesically complete strong Riemannian manifold might not be metrically complete and it might have closed and bounded subsets that are not compact.[citation needed] Further, a strong Riemannian manifold for which all closed and bounded subsets are compact might not be geodesically complete.[citation needed]
If is a weak Riemannian metric, then no notion of completeness implies the other in general.[citation needed]
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