Riemannian manifold

Last updated

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

Contents

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 (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.

Introduction

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

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

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 (Gromov 1999) and (Shi and Tam 2002).

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 (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

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 (canonical) 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.

Immersions

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

Product metrics

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 (as above), 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.

Proof 

Let M be a differentiable manifold and {(Uα, φα) | αI} a locally finite atlas of open subsets Uα of M and diffeomorphisms onto open subsets of Rn

Let {τα}αI be a differentiable partition of unity subordinate to the given atlas.

Then define the metric g on M by

where gcan is the Euclidean metric on Rn and is its pullback along φβ.

This is readily seen to be a metric on M.

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 (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

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.

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 (Gromov 1999)

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 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:

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 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.

See also

Related Research Articles

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In geometry, a geodesic is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 2. The sectional curvature Kp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a Minkowski functionalF(x,−) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ : [a,b] → M as

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.

In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces.

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

Differentiable manifold Manifold upon which it is possible to perform calculus

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems.

In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy. As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.

In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is developed in this article.

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.

In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete, this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.

References