In differential geometry, a Riemannian manifold or Riemannian space (M, g) 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 (U, x) on M, the n2 functions
The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
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.
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).
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
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.
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.
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
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.
Riemannian metrics are defined in a way similar to the finite-dimensional case. However there is a distinction between two types of Riemannian metrics:
In the case of strong Riemannian metrics, a part of the finite-dimensional Hopf-Rinow still works.