Riemannian manifold - AbsoluteAstronomy.com

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

and the differential geometry of surfaces

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold

Differentiable manifold

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...

M in which each tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

is equipped with an inner product

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

g, a Riemannian metric, which varies smoothly from point to point. The terms are named after German mathematician Bernhard Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

.

A Riemannian metric makes it possible to define various geometric notions on a Riemannian manifold, such as angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s, lengths of curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s, area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

s (or volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

s), curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

, gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

s of functions and divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

Introduction

In 1828, Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

proved his Theorema Egregium

Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...

(remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

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

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

. Bernhard Riemann

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 was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

used the theory of Riemannian manifolds to develop his General Theory of Relativity. In particular, his equations for gravitation are restrictions on the curvature of space.

Overview

The tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space

Tangent space

, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

α(t): [0, 1] → M has tangent vector α′(t₀) in the tangent space TM(α(t₀)) at any point t₀ ∈ (0, 1), and each such vector has length ‖α′(t₀)‖, where ‖·‖ denotes the norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

induced by the inner product on TM(α(t₀)). The integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of these lengths gives the length of the curve α:

Smoothness of α(t) for t in [0, 1] guarantees that the integral L(α) exists and the length of this curve is defined.

In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important.

Every smooth submanifold

Glossary of differential geometry and topology

This is a glossary of terms specific to differential geometry and differential topology.The following two glossaries are closely related:*Glossary of general topology*Glossary of Riemannian and metric geometry.See also:*List of differential geometry topics...

of Rⁿ has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on Rⁿ. In fact, as follows from the Nash embedding theorem

Nash embedding theorem

The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...

, all Riemannian manifolds can be realized this way.
In particular one could define Riemannian manifold as a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

which is isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

to a smooth submanifold of Rⁿ with the induced intrinsic metric

Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...

, where isometry here is meant in the sense of preserving the length of curves. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry

Riemannian geometry

Riemannian manifolds as metric spaces

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section

Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...

of the positive-definite quadratic forms on the tangent bundle

Tangent bundle

. Then one has to work to show that it can be turned to a metric space:

If γ: [a, b] → M is a continuously differentiable curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

in the Riemannian manifold M, then we define its length L(γ) in analogy with the example above by

With this definition of length, every connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

Riemannian manifold M becomes a metric space

Metric space

(and even a length metric space

Intrinsic metric

) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as

d(x,y) = inf

Infimum

In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

{ L(γ) : γ is a continuously differentiable curve joining x and y}.

Even though Riemannian manifolds are usually "curved," there is still a notion of "straight line" on them: the geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s. These are curves which locally join their points along shortest path

Geodesic

s.

Assuming the manifold is compact, any two points x and y can be connected with a geodesic whose length is d(x,y). Without compactness, this need not be true. For example, in the punctured plane R² \ {0}, the distance between the points (−1, 0) and (1, 0) is 2, but there is no geodesic realizing this distance.

Properties

In Riemannian manifolds, the notions of geodesic

Geodesic

completeness

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

, topological

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

completeness and metric

Metric space

completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem.

Riemannian metrics

Let M be a differentiable manifold

Differentiable manifold

of dimension n. A Riemannian metric on M is a family of (positive definite) inner products

such that, for all differentiable vector fields X,Y on M,

defines a smooth function

Smooth function

M → R.

More formally, a Riemannian metric

is a symmetric (0,2)-tensor that is positive definite (i.e. g(X,X) > 0 for all tangent vectors X ≠ 0).

In a system of local coordinates

Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...

on the manifold M given by n real-valued functions x¹,x², …, xⁿ, the vector fields

give a basis of tangent vector

Tangent space

s at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,

Equivalently, the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

can be written in terms of the dual basis {dx¹, …, dxⁿ} of the cotangent bundle as

Endowed with this metric, the differentiable manifold

Differentiable manifold

(M,g) is a Riemannian manifold.

Examples

With identified with , the standard metric over an open subset
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

is defined by

Then g is a Riemannian metric, and

Equipped with this metric, Rⁿ is called Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

of dimension n and g_ij^can is called the (canonical) Euclidean metric.

Let (M,g) be a Riemannian manifold and be a submanifold
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

of M. Then the restriction of g to vectors tangent along N defines a Riemannian metric over N.
More generally, let f:Mⁿ→N^n+k be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:

This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion.

Let (M,g^M) be a Riemannian manifold, h:M^n+k→N^k be a differentiable map and q∈N be a regular value of h (the differential dh(p) is surjective for all p∈h^-1(q)). Then h^-1(q)⊂M is a submanifold of M of dimension n. Thus h^-1(q) carries the Riemannian metric induced by inclusion.

In particular, consider the following map :

Then, 0 is a regular value of h and

is the unit sphere

. The metric induced from

is called the canonical metric of

Let and be two Riemannian manifolds and consider the cartesian product with the product structure. Furthermore, let and be the natural projections. For , a Riemannian metric on can be introduced as follows :

The identification

allows us to conclude that this defines a metric on the product space.

The torus

possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from

on the circle

and then taking the product metric. The torus

endowed with this metric is called the flat torus.

Let be two metrics on . Then,

is also a metric on M.

The pullback metric

If f:M→N is a differentiable map and (N,g^N) a Riemannian manifold, then the pullback of g^N along f is a quadratic form on the tangent space of M. The pullback is the quadratic form f*g^N on TM defined for v, w ∈ T_pM by

where df(v) is the pushforward of v by f.

The quadratic form

is in general only a semi definite form because

can have a kernel. If f is a diffeomorphism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

, or more generally an immersion, then it defines a Riemannian metric on M, the pullback metric. In particular, every embedded smooth submanifold

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold.

Existence of a metric

Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {(U_α, φ(U_α))|α∈I} a locally finite

Locally finite collection

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....

atlas

Atlas (topology)

In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

of open subsets U of M and diffeomorphisms onto open subsets of Rⁿ

Let τ_α be a differentiable partition of unity subordinate to the given atlas. Then define the metric g on M by

where g^can is the Euclidean metric. This is readily seen to be a metric on M.

Isometries

Let

and

be two Riemannian manifolds, and

be a diffeomorphism. Then, f is called an isometry, if

or pointwise

Moreover, a differentiable mapping

is called a local isometry at

if there is a neighbourhood

, such that

is a diffeomorphism satisfying the previous relation.

Riemannian manifolds as metric spaces

A connected

Connected space

Riemannian manifold carries the structure of a metric space

Metric space

whose distance function is the arclength of a minimizing geodesic

Geodesic

.

Specifically, let (M,g) be a connected Riemannian manifold. Let

be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as

By change of variables

Change of variables

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with new ones; the new and old variables being related in some specified way...

, the arclength is independent of the chosen parametrization. In particular, a curve

can be parametrized by its arc length. A curve is parametrized by arclength if and only if

for all

.

The distance function d : M×M → [0,∞) is defined by

where the infimum

Infimum

extends over all differentiable curves γ beginning at p∈M and ending at q∈M.

This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that d(p,q)=0 implies that p=q. For this property, one can use a normal coordinate system

Normal coordinates

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...

, which also allows one to show that the topology induced by d is the same as the original topology on M.

Diameter

The diameter of a Riemannian manifold M is defined by

The diameter is invariant under global isometries. Furthermore, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

if and only if it is complete and has finite diameter.

Geodesic completeness

A Riemannian manifold M is geodesically complete if for all

, the exponential map

is defined for all

, i.e. if any geodesic

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 which are not complete.