Riemannian manifold


In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.

The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds.

A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions

are smooth functions, i.e., they are infinitely differentiable. These functions are commonly designated as .

With further restrictions on the , one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities.

A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.

Introduction edit

In 1828, Carl Friedrich Gauss proved his Theorema Egregium ("remarkable theorem" in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See Differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that is intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime.

Definition edit

The tangent bundle of a smooth manifold   assigns to each point   of   a vector space   called the tangent space of   at   A Riemannian metric (by its definition) assigns to each   a positive-definite inner product   along with which comes a norm   defined by   The smooth manifold   endowed with this metric   is a Riemannian manifold, denoted  .

When given a system of smooth local coordinates on   given by   real-valued functions   the vectors


form a basis of the vector space   for any   Relative to this basis, one can define metric tensor "components" at each point   by


One could consider these as   individual functions   or as a single   matrix-valued function on   note that the "Riemannian" assumption says that it is valued in the subset consisting of symmetric positive-definite matrices.

In terms of tensor algebra, the metric tensor can be written in terms of the dual basis {dx1, ..., dxn} of the cotangent bundle as


Isometries edit

If   and   are two Riemannian manifolds, with   a diffeomorphism, then   is called an isometry if   i.e. if


for all   and  

One says that a 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).

Regularity of a Riemannian metric edit

One says that the Riemannian metric   is continuous if   are continuous when given any smooth coordinate chart   One says that   is smooth if these functions are smooth when given any smooth coordinate chart. One could also consider many other types of Riemannian metrics in this spirit.

In most expository accounts of Riemannian geometry, the metrics are always taken to be smooth. However, there can be important reasons to consider metrics which are less smooth. Riemannian metrics produced by methods of geometric analysis, in particular, can be less than smooth. See for instance (Gromov 1999) and (Shi and Tam 2002).

Overview edit

Examples of Riemannian manifolds will be discussed below. A famous theorem of John Nash states that, given any smooth Riemannian manifold   there is a (usually large) number   and an embedding   such that the pullback by   of the standard Riemannian metric on   is   Informally, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. In this sense, it is arguable 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 the 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.

Examples edit

Euclidean space edit

Let   denote the standard coordinates on   Then define   by


Phrased differently: relative to the standard coordinates, the local representation   is given by the constant value  

This is clearly a Riemannian metric, and is called the standard Riemannian structure on   It is also referred to as Euclidean space of dimension n and gijcan is also called the (canonical) Euclidean metric.

Embedded submanifolds edit

Let   be a Riemannian manifold and let   be an embedded submanifold of   which is at least   Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.

  • For example, consider   which is a smooth embedded submanifold of the Euclidean space with its standard metric. The Riemannian metric this induces on   is called the standard metric or canonical metric on  
  • There are many similar examples. For example, every ellipsoid in   has a natural Riemannian metric. The graph of a smooth function   is an embedded submanifold, and so has a natural Riemannian metric as well.

Immersions edit

Let   be a Riemannian manifold and let   be a differentiable map. Then one may consider the pullback of   via  , which is a symmetric 2-tensor on   defined by


where   is the pushforward of   by  

In this setting, generally   will not be a Riemannian metric on   since it is not positive-definite. For instance, if   is constant, then   is zero. In fact,   is a Riemannian metric if and only if   is an immersion, meaning that the linear map   is injective for each  

  • An important example occurs when   is not simply-connected, so that there is a covering map   This is an immersion, and so the universal cover of any Riemannian manifold automatically inherits a Riemannian metric. More generally, but by the same principle, any covering space of a Riemannian manifold inherits a Riemannian metric.
  • Also, an immersed submanifold of a Riemannian manifold inherits a Riemannian metric.

Product metrics edit

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

  • Considering the decomposition   one may define
  • Let   be a smooth coordinate chart on   and let   be a smooth coordinate chart on   Then   is a smooth coordinate chart on   For convenience let   denote the collection of positive-definite symmetric   real matrices. Denote the coordinate representation of   relative to   by   and denote the coordinate representation of   relative to   by   Then the local coordinate representation of   relative to   is   given by

A standard example is to consider the n-torus   define as the n-fold product   If one gives each copy of   its standard Riemannian metric, considering   as an embedded submanifold (as above), then one can consider the product Riemannian metric on   It is called a flat torus.

Convex combinations of metrics edit

Let   and   be two Riemannian metrics on   Then, for any number  


is also a Riemannian metric on   More generally, if   and   are any two positive numbers, then   is another Riemannian metric.

Every smooth manifold has a Riemannian metric edit

This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed by only using that a smooth manifold is locally Euclidean, for this result it is necessary to include in the definition of "smooth manifold" that it is Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.


Let   be a differentiable 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  .

Then define the metric   on   by


where   is the Euclidean metric on   and   is its pullback along  .

This is readily seen to be a metric on  .

The metric space structure of continuous connected Riemannian manifolds edit

The length of piecewise continuously-differentiable curves edit

If   is differentiable, then it assigns to each   a vector   in the vector space   the size of which can be measured by the norm   So   defines a nonnegative function on the interval   The length is defined as the integral of this function; however, as presented here, there is no reason to expect this function to be integrable. It is typical to suppose g to be continuous and   to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of  


is well-defined. This definition can easily be extended to define the length of any piecewise-continuously differentiable curve.

In many instances, such as in defining the Riemann curvature tensor, it is necessary to require that g has more regularity than mere continuity; this will be discussed elsewhere. For now, continuity of g will be enough to use the length defined above in order to endow M with the structure of a metric space, provided that it is connected.

The metric space structure edit

Precisely, define   by


It is mostly straightforward to check the well-definedness of the function   its symmetry property   its reflexivity property   and the triangle inequality   although there are some minor technical complications (such as verifying that any two points can be connected by a piecewise-differentiable path). It is more fundamental to understand that   ensures   and hence that   satisfies all of the axioms of a metric.

The observation that underlies the above proof, 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 then, even when g is smooth, 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.

Geodesics edit

As in the previous section, let   be a connected and continuous Riemannian manifold; consider the associated metric space   Relative to this metric space structure, one says that a path   is a unit-speed geodesic if for every   there exists an interval   which contains   and such that


Informally, one may say that one is asking for   to locally 'stretch itself out' as much as it can, subject to the (informally considered) unit-speed constraint. The idea is that if   is (piecewise) continuously differentiable and   for all   then one automatically has   by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of   So the unit-speed geodesic condition as given above is requiring   and   to be as far from one another as possible. The fact that we are only looking for curves to locally stretch themselves out is reflected by the first two examples given below; the global shape of   may force even the most innocuous geodesics to bend back and intersect themselves.

  • Consider the case that   is the circle   with its standard Riemannian metric, and   is given by   Recall that   is measured by the lengths of curves along  , not by the straight-line paths in the plane. This example also exhibits the necessity of selecting out the subinterval   since the curve   repeats back on itself in a particularly natural way.
  • Likewise, if   is the round sphere   with its standard Riemannian metric, then a unit-speed path along an equatorial circle will be a geodesic. A unit-speed path along the other latitudinal circles will not be geodesic.
  • Consider the case that   is   with its standard Riemannian metric. Then a unit-speed line such as   is a geodesic but the curve   from the first example above is not.

Note that unit-speed geodesics, as defined here, are by necessity continuous, and in fact Lipschitz, but they are not necessarily differentiable or piecewise differentiable.

The Hopf–Rinow theorem edit

As above, let   be a connected and continuous Riemannian manifold. The Hopf–Rinow theorem, in this setting, says that (Gromov 1999)

  • if the metric space   is complete (i.e. every  -Cauchy sequence converges) then
    • every closed and bounded subset of   is compact.
    • given any   and any real numbers a, b with a < b, there is a unit-speed geodesic   from   to   such that   for all  

The essence of the proof is that once the first half is established, one may directly apply the Arzelà–Ascoli theorem, in the context of the compact metric space   to a sequence of piecewise continuously-differentiable unit-speed curves from   to   whose lengths approximate   The resulting subsequential limit is the desired geodesic.

The assumed completeness of   is important. For example, consider the case that   is the punctured plane   with its standard Riemannian metric, and one takes   and   There is no unit-speed geodesic from one to the other.

The diameter edit

Let   be a connected and continuous Riemannian manifold. As with any metric space, one can define the diameter of   to be


The Hopf–Rinow theorem shows that if   is complete and has finite diameter, then 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 statement:

  • 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.

Note that, more generally, and with the same one-line proof, every compact metric space has finite diameter. However the following statement is false: "If a metric space is complete and has finite diameter, then it is compact." For an example of a complete and non-compact metric space of finite diameter, consider


with the uniform metric


So, although all of the terms in the above corollary of the Hopf–Rinow theorem involve only the metric space structure of   it is important that the metric is induced from a Riemannian structure.

Riemannian metrics edit

Geodesic completeness edit

A Riemannian manifold M is geodesically complete if for all pM, the exponential map expp is defined for all v ∈ TpM, i.e. if any geodesic γ(t) starting from p is defined for all values of the parameter tR. The Hopf–Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space.

If M is complete, then M is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds that are not complete.

Infinite-dimensional manifolds edit

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 edit

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  
  • A strong Riemannian metric on   is a weak Riemannian metric, such that   induces the topology on   Note that if   is not a Hilbert manifold then   cannot be a strong metric.

Examples edit

  • If   is a Hilbert space, then for any   one can identify   with   By setting for all     one obtains a strong Riemannian metric.
  • 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. Its tangent bundle at the identity is the set of smooth vector fields on   Let   be a volume form on   Then one can define   the   weak Riemannian metric, on   Let     Then for   and define   The   weak Riemannian metric on   induces vanishing geodesic distance, see Michor and Mumford (2005).

The metric space structure edit

Length of curves is defined in a way similar to the finite-dimensional case. The function   is defined in the same manner and is called the geodesic distance. In the finite-dimensional case, the proof that this function is a metric uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact and so this statement may fail.

  • If   is a strong Riemannian metric on  , then   separates points (hence is a metric) and induces the original topology.
  • If   is a weak Riemannian metric but not strong,   may fail to separate points or even be degenerate.

For an example of the latter, see Valentino and Daniele (2019).

The Hopf–Rinow theorem edit

In the case of strong Riemannian metrics, a part of the finite-dimensional Hopf–Rinow still works.

Theorem: Let   be a strong Riemannian manifold. Then metric completeness (in the metric  ) implies geodesic completeness (geodesics exist for all time). Proof can be found in (Lang 1999, Chapter VII, Section 6). The other statements of the finite-dimensional case may fail. An example can be found here.

If   is a weak Riemannian metric, then no notion of completeness implies the other in general.

See also edit

References edit

  • Lee, John M. (2018). Introduction to Riemannian Manifolds. Springer-Verlag. ISBN 978-3-319-91754-2.
  • do Carmo, Manfredo (1992). Riemannian geometry. Basel: Birkhäuser. ISBN 978-0-8176-3490-2.
  • Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces (Based on the 1981 French original ed.). Birkhäuser Boston, Inc., Boston, MA. ISBN 0-8176-3898-9.
  • Jost, Jürgen (2008). Riemannian Geometry and Geometric Analysis (5th ed.). Berlin: Springer-Verlag. ISBN 978-3-540-77340-5.
  • Shi, Yuguang; Tam, Luen-Fai (2002). "Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature". J. Differential Geom. 62 (1): 79–125. arXiv:math/0301047. doi:10.4310/jdg/1090425530. S2CID 13841883.
  • Lang, Serge (1999). Fundamentals of differential geometry. New York: Springer-Verlag. ISBN 978-1-4612-0541-8.
  • Magnani, Valentino; Tiberio, Daniele (2020). "A remark on vanishing geodesic distances in infinite dimensions". Proc. Amer. Math. Soc. 148 (1): 3653–3656. arXiv:1910.06430. doi:10.1090/proc/14986. S2CID 204578276.
  • Michor, Peter W.; Mumford, David (2005). "Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms". Documenta Math. 10: 217–245. arXiv:math/0409303. doi:10.4171/dm/187. S2CID 69260.

External links edit