Riemannian manifold


In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. A common convention is to take g to be smooth, which means that for any smooth coordinate chart on M, the n2 functions
are smooth functions. In the same way, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities.
The family gp of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
A Riemannian metric 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, extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.

Introduction

In 1828, Carl Friedrich Gauss proved his Theorema Egregium, 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 to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of spacetime.

Definition

The tangent bundle of a smooth manifold assigns to each point of a vector space called the tangent space of at A Riemannian metric 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 of the cotangent bundle as

Isometries

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 a diffeomorphism and isometry.

Regularity of a Riemannian metric

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 and.

Overview

Examples of Riemannian manifolds will be discussed below. A famous theorem of John Nash states that, given any smooth Riemannian manifold there is a 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

Euclidean space

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 Euclidean metric.

Embedded submanifolds

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.
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
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.
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, then one can consider the product Riemannian metric on It is called a flat torus.

Convex combinations of metrics

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

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.

The metric space structure of continuous connected Riemannian manifolds

The length of piecewise continuously-differentiable curves

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

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. It is more fundamental to understand that ensures and hence that satisfies all of the axioms of a metric.
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.

Geodesics

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 unit-speed constraint. The idea is that if is 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.
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

As above, let be a connected and continuous Riemannian manifold. The Hopf-Rinow theorem, in this setting, says that
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

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 must be compact. Conversely, if is compact, then the function must have a maximum, since it is a continuous function on a compact metric space. This proves the following statement:
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

Geodesic completeness

A Riemannian manifold M is geodesically complete if for all, the exponential map expp is defined for all, i.e. if any geodesic γ starting from p is defined for all values of the parameter. 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.