In differential geometry, a Riemannian manifold (or Riemann space) 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 take their name from German mathematician Bernhard Riemann, who first conceptualized them in 1854.
Formally, a Riemannian metric (or just a metric) on a smooth manifold is a smoothly varying 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 mathematics, 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
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). 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. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called 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. 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.
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
Riemannian metrics and Riemannian manifolds
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 such a "measuring stick" on every tangent space.
A Riemannian metric on assigns to each a positive-definite symmetric bilinear form (i.e. an inner product) in a smooth way (see the section on regularity below). This induces a norm defined by . A smooth manifold endowed with a Riemannian metric is a Riemannian manifold, denoted . 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 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 . 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
Regularity of the Riemannian metric 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
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 .
Isometries 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 , 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).
Volume An oriented -dimensional Riemannian manifold has a unique -form called the Riemannian volume form. The Riemannian volume form is preserved by orientation-preserving isometries. The volume form gives rise to a measure on which allows measurable functions to be integrated. If is compact, the volume of is .
Examples Euclidean space
Let denote the standard coordinates on The (canonical) Euclidean metric is given by
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
Let be a Riemannian manifold and let be an immersed submanifold or an embedded submanifold of . The pullback of is a Riemannian metric on , called the induced metric, and is said to be a Riemannian submanifold of .
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 . 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.
Products
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
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
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 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 Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric.
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.
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.