Riemannian manifold - Citizendia

In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. Elliptic geometry is also sometimes called Riemannian geometry. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. This allows one to define various notions such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Hilbert space. This article assumes some familiarity with Analytic geometry and the concept of a limit. The terms are named after German mathematician Bernhard Riemann.

1 Overview
- 1.1 Riemannian manifolds as metric spaces
- 1.2 Properties
2 Riemannian metrics
3 Riemannian manifolds as metric spaces
- 3.1 Diameter
- 3.2 Geodesic completeness
4 See also
5 External links
6 References

Overview

The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since 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 α(t): [0, 1] → M has tangent vector α′(t₀) in the tangent space TM(t₀) at any point t₀ ∈ (0, 1), and each such vector has length ||α′(t₀)||, where ||·|| denotes the norm induced by the inner product on TM(t₀). In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length The integral of these lengths gives the length of the curve α:

$L(\alpha) = \int_0^1{\|\alpha^{\prime}(t)\|\, \mathrm{d}t}.$

In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Every smooth submanifold of Rⁿ has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on Rⁿ. This is a Glossary of terms specific to Differential geometry and Differential topology. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. The Nash embedding theorems (or imbedding theorems) named after John Forbes Nash, state that every n-dimensional Riemannian manifold can be isometrically In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of Rⁿ with the induced intrinsic metric, where isometry here is meant in the sense of preserving the length of curves. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined For the Mechanical engineering and Architecture usage see Isometric projection. In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry. Elliptic geometry is also sometimes called Riemannian geometry.

Riemannian manifolds as metric spaces

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the tangent bundle. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the 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 in the Riemannian manifold M, then we define its length L(γ) in analogy with the example above by

$L(\gamma) = \int_a^b \|\gamma'(t)\|\, \mathrm{d}t.$

With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as

d(x,y) = inf{ L(γ) : γ is a continuously differentiable curve joining x and y}. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of

Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces These are curves which locally join their points along shortest paths. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces

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. This is a glossary of some terms used in the branch of Mathematics known as Topology.

Properties

In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, the Hopf–Rinow theorem is a set of statements about the Geodesic completeness of Riemannian manifolds It is named after

Riemannian metrics

Let M be a second countable Hausdorff differentiable manifold of dimension n. In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. A Riemannian metric on M is a family of inner products

$g_p : T_pM\times T_pM\longrightarrow \mathbb R,\qquad p\in M$

such that, for all differentiable vector fields $X,Y\in\mathcal V(M)$ , the application

$M\longrightarrow \mathbb R,\qquad p\longmapsto g_p(X(p), Y(p))$

is differentiable. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. Let $\{\left(\frac{\partial }{\partial x_i}\right)_p\}_i$ be a basis of tangent vectors over $p\in M$ . Then, the coefficients

$g_{ij}(p):=\Big\langle\left(\frac{\partial }{\partial x_i}\right)_p,\left(\frac{\partial }{\partial x_j}\right)_p\Big\rangle_p$

give rise to the metric tensor of rank 2

$g:=\sum_{i,j}g_{ij}\mathrm d x_i\otimes \mathrm d x_j.$

Endowed with this metric, the differentiable manifold $(M, g)$ is a Riemannian manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

Examples

With $\frac{\partial }{\partial x_i}$ identified with $e_i=(0,\dots, 1,\dots, 0)$ , the standard metric over an open subset $U\subset\mathbb R^n$ is defined by

$g^{\mathrm{can}}_p : T_pU\times T_pU\longrightarrow \mathbb R,\qquad \left(\sum_ia_i\frac{\partial}{\partial x_i},\sum_jb_j\frac{\partial}{\partial x_j}\right)\longmapsto \sum_i a_ib_i.$

Then g is a Riemannian metric, and

$g^{\mathrm{can}}_{ij}=\langle e_i,e_j\rangle = \delta_{ij}.$

Equipped with this metric, Rⁿ is called Euclidean space of dimension n and g_ij^can is called the Euclidean metric. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler

Let (M,g) be a Riemannian manifold and $N\subset M$ be a submanifold of M. 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 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. In Mathematics, an immersion is a Differentiable map between Differentiable manifolds whose derivative is everywhere Injective. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:

$g^M_p : T_pM\times T_pM\longrightarrow \mathbb R,\qquad (u,v)\longmapsto g^M_p(u,v):=g^N_{f(p)}(T_pf(u), T_pf(v)).$

This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from

Let (M,g^M) be a Riemannian manifold, h:M^n+k→N^k be a differentiable application and q∈N be a regular value of h (the differential dh(p) is surjective for all p∈h^-1(q)). In Mathematics, a critical point (or critical number) is a point on the domain of a function where one dimension Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Then N=h^-1(q)⊂M is a submanifold of M of dimension n. Thus N carries the Riemannian metric induced by inclusion.

In particular, consider the following application :

$h: \mathbb R^n\longrightarrow \mathbb R,\qquad (x_1, \dots, x_n)\longmapsto \sum_{i=1}^nx_i^2-1.$

Then, 0 is a regular value of h and

$h^{-1}(0)=\{x\in\mathbb R^n\vert \sum_{i=1}^nx_i^2=1\}=S^{n-1}$

is the unit sphere $S^{n-1}\subset \mathbb R^n$ . The metric induced from $\mathbb R^n$ on

S n - 1

is called the canonical metric of

S n - 1

Let $M 1$ and $M 2$ be two Riemannian manifolds and consider the cartesian product $M_1\times M_2$ with the product structure. Furthermore, let $\pi_1:M_1\times M_2\rightarrow M_1$ and $\pi_2:M_1\times M_2\rightarrow M_2$ be the natural projections. For $(p,q)\in M_1\times M_2$ , a Riemannian metric on $M_1\times M_2$ can be introduced as follows :

$g^{M_1\times M_2}_{(p,q)}:T_{(p,q)}(M_1\times M_2)\times T_{(p,q)}(M_1\times M_2) \longrightarrow \mathbb R,\qquad (u,v)\longmapsto g^{M_1}_p(T_{(p,q)}\pi_1(u), T_{(p,q)}\pi_1(v))+g^{M_2}_q(T_{(p,q)}\pi_2(u), T_{(p,q)}\pi_2(v)).$

The identification

$T_{(p,q)}(M_1\times M_2) \cong T_pM_1\oplus T_qM_2$

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

The torus $S^1\times\dots \times S^1=T^n$ possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from $\mathbb R^2$ on the circle $S^1\subset \mathbb R^2$ and then taking the product metric. The torus

T n

endowed with this metric is called the flat torus.

Let $g 0, g 1$ be two metrics on $M$ . Then,

$\tilde g:=\lambda g_0 + (1-\lambda)g_1,\qquad \lambda\in [0,1],$

is also a metric on M.

The pullback metric

If f:M→N is a diffeomorphism and (N,g^N) be a Riemannian manifold, then the pullback of g^N along f is a Riemannian metric on M. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from The pullback is the metric f*g^N on M defined for v, w ∈ T_pM by

(f * g N)(v, w) = g N (d f (v), d f (w)).

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

$\phi : U_\alpha\to \phi(U_\alpha)\subseteq\mathbb{R}^n.$

Let τ_α be a differentiable partition of unity subordinate to the given atlas. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X Then define the metric g on M by

$g:=\sum_\beta\tau_\beta\cdot\tilde{g}_\beta,\qquad\text{with}\qquad\tilde{g}_\beta:=\tilde{\phi}_\beta^*g^{\mathrm{can}}.$

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

Isometries

Let $(M, g M)$ and $(N, g N)$ be two Riemannian manifolds, and $f:M\rightarrow N$ be a diffeomorphism. Then, f is called an isometry, if

$g^M_p(u,v) = g^N_{f(p)}(T_pf(u), T_pf(v))\qquad \forall p\in M, \forall u,v\in T_pM.$

Moreover, a differentiable mapping $f:M\rightarrow N$ is called a local isometry at $p\in M$ if there is a neighbourhood $U\subset M$ , $U\ni p$ , such that $f:U\rightarrow f(U)$ is a diffeomorphism satisfying the previous relation.

Riemannian manifolds as metric spaces

A connected Riemannian manifold carries the structure of a metric space whose distance function is the arclength of a minimizing geodesic. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces

Specifically, let (M,g) be a connected Riemannian manifold. Let $c:[a,b]\rightarrow M$ be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as

$L_a^b(c) := \int_a^b \sqrt{g(c'(t),c'(t))}\,\mathrm d t = \int_a^b\|c'(t)\|\,\mathrm d t$

By change of variables, the arclength is independent of the chosen parametrization. In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In particular, a curve $[a,b]\rightarrow M$ can be parametrized by its arc length. A curve is parametrized by arclength if and only if $\|c'(t)\|=1$ for all $t\in[a,b]$ .

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

$d(p,q) = \inf L(\gamma)$

where the infimum extends over all differentiable curves γ beginning at p∈M and ending at q∈M. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of

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, which also allows one to show that the topology induced by d is the same as the original topology on M. In Riemannian geometry, the normal coordinates at p consist of a chart such that locally the symmetric part of the Christoffel symbols vanish i

Diameter

The diameter of a Riemannian manifold M is defined by

$\mathrm{diam}(M):=\sup_{p,q\in M} d(p,q)\in \mathbb R_{\geq 0}\cup\{+\infty\}.$

The diameter is invariant under global isometries. Furthermore, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact if and only if it is complete and has finite diameter. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has

Geodesic completeness

A Riemannian manifold M' is geodesically complete if for all $p\in M$ , the exponential map $exp p$ is defined for all $v\in T_pM$ , i. e. if any geodesic $γ(t)$ starting from p is defined for all values of the parameter $t\in\mathbb R$ . The Hopf-Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space. In Mathematics, the Hopf–Rinow theorem is a set of statements about the Geodesic completeness of Riemannian manifolds It is named after In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has

If M is complete, then M is non-extendable in the sense that it is not isometric to a proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete.

External links

L. Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over In Mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space A. Sidorov (2001), “Riemannian metric”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

References

Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis (3rd ed. The Encyclopaedia of Mathematics is a large reference work in Mathematics. ), Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1
do Carmo, Manfredo (1992), Riemannian geometry, Basel, Boston, Berlin: Birkhäuser, ISBN 978-0-8176-3490-2 [1]

Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic

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Contents

Overview

Riemannian manifolds as metric spaces

Properties

Riemannian metrics

Examples

The pullback metric

Existence of a metric

Isometries

Riemannian manifolds as metric spaces

Diameter

Geodesic completeness

See also

External links

References