Riemannian manifold

(Redirected from Riemann space)

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.

The dot product of two vectors tangent to the sphere sitting inside 3-dimensional Euclidean space contains information about the lengths and angle between the vectors. The dot products on every tangent plane, packaged together into one mathematical object, are a Riemannian metric.

Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential 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 and parallel transport.

Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the ambient space. The same is true for any submanifold of Euclidean space of any dimension. Although John 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 and projective space, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics on Lie groups and homogeneous spaces are defined intrinsically by using group actions to transport an inner product on a single tangent space to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using tools from partial differential equations.

Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of math, including geometric topology, complex geometry, and algebraic geometry. Applications include physics (especially general relativity and gauge theory), computer graphics, machine learning, and cartography. Generalizations of Riemannian manifolds include pseudo-Riemannian manifolds, Finsler manifolds, and sub-Riemannian manifolds.

History

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Riemannian manifolds were first conceptualized by their namesake, German mathematician Bernhard Riemann.

In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form).[1] This result is known as the Theorema Egregium ("remarkable theorem" in Latin).

A map that preserves the local measurements of a surface is called a local isometry. Call a property of a surface an intrinsic property if it is preserved by local isometries and call it 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 by Bernhard Riemann in 1854.[2] However, they would not be formalized until much later. In fact, the more primitive concept of a smooth manifold was first explicitly defined only in 1913 in a book by Hermann Weyl.[2]

Élie Cartan introduced the Cartan connection, one of the first concepts of a connection. Levi-Civita defined the Levi-Civita connection, a special connection on a Riemannian manifold.

Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop general relativity. Specifically, the Einstein field equations are constraints on the curvature of spacetime, which is a 4-dimensional pseudo-Riemannian manifold.

Definition

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Riemannian metrics and Riemannian manifolds

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A tangent plane of the sphere with two vectors in it. A Riemannian metric allows one to take the inner product of these vectors.

Let   be a smooth manifold. For each point  , there is an associated vector space   called the tangent space of   at  . Vectors in   are thought of as the vectors tangent to   at  .

However,   does not come equipped with an inner 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 a measuring stick on every tangent space.

A Riemannian metric   on   assigns to each   a positive-definite 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 a Riemannian manifold, denoted  .[3] A Riemannian metric is a special case of a metric tensor.

A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.

The Riemannian metric in coordinates

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If   are smooth local 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

 .[4]

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 a symmetric positive-definite matrix at  .

In terms of the tensor algebra, the Riemannian metric can be written in terms of the dual basis   of the cotangent bundle as

 [4]

Regularity of the Riemannian metric

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The Riemannian metric   is continuous if its components   are continuous in any smooth coordinate chart   The Riemannian metric   is smooth if its components   are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as Lipschitz Riemannian metrics or measurable Riemannian metrics.

There are situations in geometric 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.

Musical isomorphism

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In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its dual given by  , a Riemannian metric induces an isomorphism of bundles between the tangent bundle and the cotangent bundle. Namely, if   is a Riemannian metric, then

 

is a isomorphism of smooth vector bundles from the tangent bundle   to the cotangent bundle  .[5]

Isometries

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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 called isometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.

Specifically, if   and   are two Riemannian manifolds, a diffeomorphism   is called an isometry 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 a local isometry if every   has an open neighborhood   such that   is an isometry (and thus a diffeomorphism).[6]

Volume

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An oriented  -dimensional Riemannian manifold   has a unique  -form   called the Riemannian volume form.[7] The Riemannian volume form is preserved by orientation-preserving isometries.[8] The volume form gives rise to a measure on   which allows measurable functions to be integrated.[citation needed] If   is compact, the volume of   is  .[7]

Examples

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Euclidean space

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Let   denote the standard coordinates on   The (canonical) Euclidean metric   is given by[9]

 

or equivalently

 

or equivalently by its coordinate functions

  where   is the Kronecker delta

which together form the matrix

 

The Riemannian manifold   is called Euclidean space.

Submanifolds

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The  -sphere   with the round metric is an embedded Riemannian submanifold of  .

Let   be a Riemannian manifold and let   be an immersed submanifold or an embedded submanifold of  . The pullback   of   is a Riemannian metric on  , and   is said to be a Riemannian 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 the pushforward of   by  

Examples:

  • The  -sphere
     
is a smooth embedded submanifold of Euclidean space  .[11] The Riemannian metric this induces on   is called the round metric or standard metric.
  • Fix real numbers  . The ellipsoid
     
is a smooth embedded submanifold of Euclidean space  .
  • The graph of a smooth function   is a smooth embedded submanifold of   with its standard metric.
  • If   is not simply connected, there is a covering map  , where   is the universal cover of  . This is an immersion (since it is locally a diffeomorphism), so   automatically inherits a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.

On the other hand, if   already has a Riemannian metric  , then the immersion (or embedding)   is called an isometric immersion (or isometric embedding) if  . Hence isometric immersions and isometric embeddings are Riemannian submanifolds.[10]

Products

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A torus naturally carries a Euclidean metric, obtained by identifying opposite sides of a square (left). The resulting Riemannian manifold, called a flat torus, cannot be isometrically embedded in 3-dimensional Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.

Let   and   be two Riemannian manifolds, and consider the product manifold  . The Riemannian metrics   and   naturally put a Riemannian metric   on   which can be described in a few ways.

  • Considering the decomposition   one may define
     [12]
  • If   is a smooth coordinate chart on   and   is a smooth coordinate chart on  , then   is a smooth coordinate chart on   Let   be the representation of   in the chart   and let   be the representation of   in the chart  . The representation of   in the coordinates   is
      where  [12]

For example, consider the  -torus  . If each copy of   is given the round metric, the product Riemannian manifold   is called the flat torus. As another example, the Riemannian product  , where each copy of   has the Euclidean metric, is isometric to   with the Euclidean metric.

Positive combinations of metrics

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Let   be Riemannian metrics on   If   are any positive smooth functions on  , then   is another Riemannian metric on  

Every smooth manifold admits a Riemannian metric

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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 are Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.

Proof that every smooth manifold admits a Riemannian metric

Let   be a smooth manifold and   a locally finite atlas so that   are open subsets and   are diffeomorphisms. Such an atlas exists because the manifold is paracompact.

Let   be a differentiable partition 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 its pullback 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 the Whitney embedding theorem to embed   into Euclidean space and then pulls back the metric from Euclidean space to  . On the other hand, the Nash embedding theorem states that, given any smooth Riemannian manifold   there is an embedding   for some   such that the pullback 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 the set 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.

Metric space structure

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An admissible 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 a metric space, and the metric 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  

In verifying that   satisfies all of the axioms 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 around p which every curve from p to q 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 around p, any curve from p to q must first pass though a certain "inner radius." The assumed continuity of the Riemannian metric g 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. Let R denote  . Now, given any admissible curve   from p to q, 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 to R. So we conclude  

The observation about comparison between lengths measured by g 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 a metric space and the metric topology on   coincides with the topology on  .[15]

Diameter

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The diameter of the metric space   is

 

The Hopf–Rinow theorem shows that if   is complete 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.

If   is complete, then it is compact if and only if it has finite diameter.

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 that any complete metric space of finite diameter must be compact; it matters that the metric space came from a Riemannian manifold.

Connections, geodesics, and curvature

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Connections

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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 of vector fields on  . An (affine) connection

 

on   is a bilinear map   such that

  1. For every function  ,  
  2. The product rule   holds.[16]

The expression   is called the covariant derivative of   with respect to  .

Levi-Civita connection

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Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the Levi-Civita connection.

A connection   is said to preserve the metric if

 

A connection   is torsion-free if

 

where   is the Lie bracket.

A Levi-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  .

Covariant derivative along a curve

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If   is a smooth curve, a smooth 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 and scalar multiplication.[18] One can also pointwise multiply a smooth vector field along   by a smooth function  :

  for  

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 an extension of  .

Given a fixed connection   on   and a smooth curve  , there is a unique operator  , called the covariant derivative along  , such that:[19]

  1.  
  2.  
  3. If   is an extension of  , then  .

Geodesics

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In Euclidean space   (left), the maximal geodesics are straight lines. In the round sphere   (right), the maximal geodesics are great circles.

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. The acceleration of   is the vector field   along  . If   for all  ,   is called a geodesic.[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 a maximal 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]

Examples

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  • The nonconstant maximal geodesics of the Euclidean plane   are exactly the straight lines.[22] This agrees the fact from Euclidean geometry that the shortest path between two points is a straight line segment.
  • The nonconstant maximal geodesics of   with the round metric are exactly the great circles.[24] Since the Earth is approximately a sphere, this means that the shortest path a plane can fly between two locations on Earth is a segment of a great circle.

Hopf–Rinow theorem

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The punctured plane   is not geodesically complete because the maximal geodesic with initial conditions  ,   does not have domain  .

The Riemannian manifold   with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is  .[25] The plane   is geodesically complete. On the other hand, the punctured plane   with the restriction of the Riemannian metric from   is not geodesically complete as the maximal geodesic with initial conditions  ,   does not have domain  .

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let   be a connected Riemannian manifold. The following are equivalent:[26]

  • The metric space   is complete (every  -Cauchy sequence converges),
  • All closed and bounded subsets of   are compact,
  •   is geodesically complete.

Parallel transport

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Parallel transport of a tangent vector along a curve in the sphere.

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 curve   parallel 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 the punctured plane  . The curve the parallel transport is done along is the unit circle. In polar 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.

Parallel transports on the punctured plane under Levi-Civita connections
This transport is given by the metric  .
This transport is given by the metric  .

Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

Riemann curvature tensor

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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  . The Riemann curvature tensor is the map   defined by

 

where   is the Lie bracket of vector fields. The Riemann curvature tensor is a  -tensor field.[30]

Ricci curvature tensor

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Fix a connection   on  . The Ricci curvature tensor is

 

where   is the trace. The Ricci curvature tensor is a covariant 2-tensor field.[31]

Einstein manifolds

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The Ricci curvature tensor   plays a defining role in the theory of Einstein manifolds, which has applications to the study of gravity. A (pseudo-)Riemannian metric   is called an Einstein metric if Einstein's equation

  for some constant  

holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called an Einstein manifold.[32] Examples of Einstein manifolds include Euclidean space, the  -sphere, hyperbolic space, and complex projective space with the Fubini-Study metric.

Scalar curvature

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Constant curvature and space forms

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A Riemannian manifold is said to have constant curvature κ if every sectional curvature equals the number κ. This is equivalent to the condition that, relative to any coordinate chart, the Riemann curvature tensor can be expressed in terms of the metric tensor as

 

This implies that the Ricci curvature is given by Rjk = (n – 1)κgjk and the scalar curvature is n(n – 1)κ, where n is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is an Einstein manifold, thereby having constant scalar curvature. As found by Bernhard Riemann in his 1854 lecture introducing Riemannian geometry, the locally-defined Riemannian metric

 

has constant curvature κ. Any two Riemannian manifolds of the same constant curvature are locally isometric, and so it follows that any Riemannian manifold of constant curvature κ can be covered by coordinate charts relative to which the metric has the above form.[33]

A Riemannian space form is a Riemannian manifold with constant curvature which is additionally connected and geodesically complete. A Riemannian space form is said to be a spherical space form if the curvature is positive, a Euclidean space form if the curvature is zero, and a hyperbolic space form or hyperbolic 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 curvature 1, 0, and –1 respectively. Furthermore, the Killing–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 the covering manifold construction, any Riemannian space form is isometric to the quotient manifold of a simply-connected Riemannian space form, modulo a certain group action of isometries. For example, the isometry group of the n-sphere is the orthogonal group O(n + 1). Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue, the natural group action of the orthogonal group on the n-sphere restricts to a group action of G, with the quotient manifold Sn / G inheriting a geodesically complete Riemannian metric of constant curvature 1. Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in group 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 or real 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 the lens spaces and the Poincaré 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 the Klein bottle, the Möbius strip, the torus, the cylinder S1 × 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 with Teichmüller 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 as hyperbolic geometry.[35]

Riemannian metrics on Lie groups

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Left-invariant metrics on Lie groups

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Let G be a Lie group, such as the group 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 product ge on the tangent space at the identity, the inner product on the tangent space at an arbitrary point p is defined by

 

where for arbitrary x, Lx is the left multiplication map GG sending a point y to xy. Riemannian metrics constructed this way are left-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 of ge, the adjoint representation of G, and the Lie algebra associated to G.[36] These formulas simplify considerably in the special case of a Riemannian metric which is bi-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 the special unitary group SU(2), are among the simplest examples of the collapsing 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 not Einstein, 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 a compact Lie group with an abelian Lie group.[42]

Homogeneous spaces

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A Riemannian manifold (M, g) is said to be homogeneous if for every pair of points x and y in M, there is some isometry f of the Riemannian manifold sending x to y. This can be rephrased in the language of group actions as the requirement that the natural action of the isometry 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 group G with compact subgroup K which does not contain any nontrivial normal subgroup of G, fix any complemented subspace W of the Lie algebra of K within the Lie algebra of G. If this subspace is invariant under the linear map adG(k): WW for any element k of K, then G-invariant Riemannian metrics on the coset space G/K are in one-to-one correspondence with those inner products on W which are invariant under adG(k): WW for every element k of K.[44] Each such Riemannian metric is homogeneous, with G 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 when K 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 on W, the Lie algebra of G, and the direct sum decomposition of the Lie algebra of G into the Lie algebra of K and W.[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.

Symmetric spaces

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A connected Riemannian manifold (M, g) is said to be symmetric if for every point p of M there exists some isometry of the manifold with p as a fixed point and for which the negation of the differential at p is the identity 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 the Riemann curvature tensor and Ricci curvature are parallel. Riemannian manifolds with this curvature property, which could loosely be phrased as "constant Riemann curvature tensor" (not to be confused with constant curvature), are said to be locally symmetric. This property nearly characterizes symmetric spaces; Élie Cartan proved in the 1920s that a locally symmetric Riemannian manifold which is geodesically complete and simply-connected must in fact be symmetric.[45]

Many of the fundamental examples of Riemannian manifolds are symmetric. The most basic include the sphere and real projective spaces with their standard metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective space which are also symmetric, as are complex 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 are irreducible, referring to those which cannot be locally decomposed as product spaces. Every such space is an example of an Einstein manifold; among them only the one-dimensional manifolds have zero scalar curvature. These spaces are important from the perspective of Riemannian holonomy. As found in the 1950s by Marcel 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 of Kähler geometry, quaternion-Kähler geometry, G2 geometry, and Spin(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]

Infinite-dimensional manifolds

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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 a topological vector space; for example, Fréchet, Banach, and Hilbert manifolds.

Definitions

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Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics:

  • A weak Riemannian metric on   is a smooth function   such that for any   the restriction   is an inner product on  [citation needed]
  • A strong Riemannian metric on   is a weak Riemannian metric such that   induces the topology on  . If   is a strong Riemannian metric, then   must be a Hilbert manifold.[citation needed]

Examples

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  • If   is a Hilbert space, then for any   one can identify   with   The metric   for all   is a strong Riemannian metric.[citation needed]
  • Let   be a compact Riemannian manifold and denote by   its diffeomorphism group. The latter is a smooth manifold (see here) and in fact, a Lie group.[citation needed] Its tangent bundle at the identity is the set of smooth vector fields on  [citation needed] Let   be a volume form on   The   weak Riemannian metric on  , denoted  , is defined as follows. Let     Then for  ,
     .[citation needed]

Metric space structure

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Length of curves and the Riemannian distance function   are defined in a way similar to the finite-dimensional case. The distance function  , called the geodesic distance, is always a pseudometric (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-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.

  • If   is a strong Riemannian metric on  , then   separates points (hence is a metric) and induces the original topology.[citation needed]
  • If   is a weak Riemannian metric,   may fail to separate points. In fact, it may even be identically 0.[46] For example, if   is a compact Riemannian manifold, then the   weak Riemannian metric on   induces vanishing geodesic distance.[47]

Hopf–Rinow theorem

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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]

See also

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References

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Notes

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  1. ^ do Carmo 1992, pp. 35–36.
  2. ^ a b do Carmo 1992, p. 37.
  3. ^ a b do Carmo 1992, p. 38.
  4. ^ a b Lee 2018, p. 13.
  5. ^ Lee 2018, p. 26.
  6. ^ a b Lee 2018, p. 12.
  7. ^ a b Lee 2018, p. 30.
  8. ^ Lee 2018, p. 31.
  9. ^ Lee 2018, pp. 12–13.
  10. ^ a b Lee 2018, p. 15.
  11. ^ Lee 2018, p. 16.
  12. ^ a b Lee 2018, p. 20.
  13. ^ Lee 2018, p. 11.
  14. ^ Lee 2018, p. 39.
  15. ^ Burtscher 2015, p. 276.
  16. ^ Lee 2018, pp. 89–91.
  17. ^ Lee 2018, pp. 122–123.
  18. ^ Lee 2018, p. 100.
  19. ^ Lee 2018, pp. 101–102.
  20. ^ Lee 2018, p. 103.
  21. ^ Lee 2018, pp. 103–104.
  22. ^ a b c Lee 2018, p. 105.
  23. ^ Lee 2018, p. 156.
  24. ^ Lee 2018, p. 137.
  25. ^ Lee 2018, p. 131.
  26. ^ do Carmo 1992, pp. 146–147.
  27. ^ Lee 2018, pp. 105–110.
  28. ^ Lee 2018, p. 201.
  29. ^ Lee 2018, p. 200.
  30. ^ Lee 2018, pp. 196–197.
  31. ^ Lee 2018, p. 207.
  32. ^ Lee 2018, p. 210.
  33. ^ a b Wolf 2011, Chapter 2.
  34. ^ Wolf 2011, Chapters 2 and 7.
  35. ^ Wolf 2011, Chapters 2 and 3.
  36. ^ Cheeger & Ebin 2008, Proposition 3.18.
  37. ^ Cheeger & Ebin 2008, Corollary 3.19; Petersen 2016, Section 4.4.
  38. ^ Petersen 2016, Section 4.4.3 and p. 399.
  39. ^ Petersen 2016, p. 369.
  40. ^ In the upper half-space model of hyperbolic space, the Lie group structure is defined by  
  41. ^ Lee 2018, Example 3.16f.
  42. ^ Lee 2018, p. 72; Milnor 1976.
  43. ^ Kobayashi & Nomizu 1963, Theorem IV.4.5.
  44. ^ a b Besse 1987, Section 7C.
  45. ^ a b c Petersen 2016, Chapter 10.
  46. ^ a b Magnani & Tiberio 2020.
  47. ^ Michor & Mumford 2005.

Sources

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