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.
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.
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.
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 metricEdit
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).
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.
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.
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.
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
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.
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 metricsEdit
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 metricEdit
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 finiteatlas so that are open subsets and are diffeomorphisms.
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 manifoldsEdit
The length of piecewise continuously-differentiable curvesEdit
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 structureEdit
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.
(Sketched) Proof that implies
Briefly: 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 to be used at the final step of the proof.
Now, given any piecewise continuously-differentiable path 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 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.
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 theoremEdit
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 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.
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
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.
A Riemannian manifold M is geodesically complete if for all p ∈ M, 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 t ∈ R. 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.
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.
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.
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. It 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 structureEdit
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 theoremEdit
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.