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In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M, g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X|_p, Y|_p) is a smooth function. The family g_p 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 insection, 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 was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds (formally pseudo-Riemannian manifolds) to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of space-time.

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. 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 : [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 induced by the inner product on TM(α(t₀)). The integral of these lengths gives the length of the curve α:

L(α)=∫01‖α′(t)‖dt.{displaystyle L(alpha )=int _{0}^{1}|alpha '(t)|,mathrm {d} t.}

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 of Rⁿ with a Euclidean metric 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, all Riemannian manifolds can be realized this way.
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. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in 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. This definition allows the construction of an accompanying 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(γ)=∫ab‖γ′(t)‖dt.{displaystyle 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 a continuously differentiable curve joining x and y}.{displaystyle d(x,y)=inf{L(gamma ):gamma {text{ a continuously differentiable curve joining }}x{text{ and }}y}.}

Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths.

Assuming the manifold is complete, any two points x and y can be connected with a geodesic whose length is d(x, y). Without completeness, 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 completeness and 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 of dimension n. A Riemannian metric on M is a family of (positive-definite) inner products

gp:TpM×TpM→R,p∈M{displaystyle g_{p}colon T_{p}Mtimes T_{p}Mrightarrow mathbf {R} ,qquad pin M}

such that, for every pair of differentiable vector fields X, Y on M,

p↦gp(X|p,Y|p){displaystyle pmapsto g_{p}(X|_{p},Y|_{p})}

defines a smooth function M → R.

We can also regard a Riemannian metric g as a symmetric (0, 2)-tensor field that is positive-definite at every point (i.e. g(X, X)|_p := g_p(X|_p, X|_p) > 0 whenever X|_p ≠ 0).

In a system of local coordinates on the manifold M given by n real-valued functions x¹, x², ..., xⁿ, the vector fields

{∂∂x1,…,∂∂xn}{displaystyle left{{frac {partial }{partial x^{1}}},dotsc ,{frac {partial }{partial x^{n}}}right}}

give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,

gij|p:=gp(∂∂xi|p,∂∂xj|p).{displaystyle g_{ij}|_{p}:=g_{p}left(left.{frac {partial }{partial x^{i}}}right|_{p},left.{frac {partial }{partial x^{j}}}right|_{p}right).}

Equivalently, the metric tensor can be written in terms of the dual basis {dx¹, ..., dxⁿ} of the cotangent bundle as

g=∑i,jgijdxi⊗dxj.{displaystyle g=sum _{i,j}g_{ij},mathrm {d} x^{i}otimes mathrm {d} x^{j}.}

The differentiable manifold M endowed with this metric g is a Riemannian manifold, denoted (M, g).

Examples

• With ∂∂xi{displaystyle {frac {partial }{partial x^{i}}}} identified with e_i = (0, ..., 1, ..., 0), the standard metric over an open subset U ⊂ Rⁿ is defined by

gpcan:TpU×TpU⟶R,(∑iai∂∂xi,∑jbj∂∂xj)⟼∑iaibi.{displaystyle g_{p}^{mathrm {can} }colon T_{p}Utimes T_{p}Ulongrightarrow mathbf {R} ,qquad left(sum _{i}a_{i}{frac {partial }{partial x^{i}}},sum _{j}b_{j}{frac {partial }{partial x^{j}}}right)longmapsto sum _{i}a_{i}b_{i}.}

gijcan=⟨ei,ej⟩=δij.{displaystyle g_{ij}^{mathrm {can} }=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 (canonical) Euclidean metric.

• Let (M, g) be a and N ⊂ M be a submanifold 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:

gpM:TpM×TpM⟶R,{displaystyle g_{p}^{M}colon T_{p}Mtimes T_{p}Mlongrightarrow mathbf {R} ,}

(u,v)⟼gpM(u,v):=gf(p)N(dfp(u),dfp(v)).{displaystyle (u,v)longmapsto g_{p}^{M}(u,v):=g_{f(p)}^{N}(df_{p}(u),df_{p}(v)).}

This is then a metric; the positive definiteness follows on 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 :

h:Rn⟶R,(x1,…,xn)⟼∑i=1n(xi)2−1.{displaystyle hcolon mathbf {R} ^{n}longrightarrow mathbf {R} ,qquad (x^{1},dotsc ,x^{n})longmapsto sum _{i=1}^{n}(x^{i})^{2}-1.}

Then, 0 is a regular value of h and

h−1(0)={x∈Rn|∑i=1n(xi)2=1}=Sn−1{displaystyle h^{-1}(0)=left{xin mathbf {R} ^{n},leftvert ,sum _{i=1}^{n}(x^{i})^{2}=1right.right}=mathbf {S} ^{n-1}}

is the unit sphere Sⁿ⁻¹ ⊂ Rⁿ. The metric induced from Rⁿ on Sⁿ⁻¹ is called the canonical metric of Sⁿ⁻¹.

• Let M₁ and M₂ be two Riemannian manifolds and consider the cartesian product M₁ × M₂ with the product structure. Furthermore, let π₁ : M₁ × M₂ → M₁ and π₂ : M₁ × M₂ → M₂ be the natural projections. For (p, q) ∈ M₁ × M₂, a Riemannian metric on M₁ × M₂ can be introduced as follows :

g(p,q)M1×M2:T(p,q)(M1×M2)×T(p,q)(M1×M2)⟶R,{displaystyle g_{(p,q)}^{M_{1}times M_{2}}colon T_{(p,q)}(M_{1}times M_{2})times T_{(p,q)}(M_{1}times M_{2})longrightarrow mathbf {R} ,}

(u,v)⟼gpM1(T(p,q)π1(u),T(p,q)π1(v))+gqM2(T(p,q)π2(u),T(p,q)π2(v)).{displaystyle (u,v)longmapsto g_{p}^{M_{1}}(T_{(p,q)}pi _{1}(u),T_{(p,q)}pi _{1}(v))+g_{q}^{M_{2}}(T_{(p,q)}pi _{2}(u),T_{(p,q)}pi _{2}(v)).}

The identification

T(p,q)(M1×M2)≅TpM1⊕TqM2{displaystyle T_{(p,q)}(M_{1}times M_{2})cong T_{p}M_{1}oplus T_{q}M_{2}}

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

The torus S¹ × ... × S¹ = Tⁿ possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from R² on the circle S¹ ⊂ R² and then taking the product metric. The torus Tⁿ endowed with this metric is called the flat torus.

• Let g₀, g₁ be two metrics on M. Then,

g~:=λg0+(1−λ)g1,λ∈[0,1],{displaystyle {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 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

(f∗gN)(v,w)=gN(df(v),df(w)).{displaystyle (f^{*}g^{N})(v,w)=g^{N}(df(v),df(w)),.}

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

The quadratic form f^∗g^N is in general only a semidefinite form because df can have a kernel. If f is a diffeomorphism, or more generally an immersion, then it defines a Riemannian metric on M, the pullback metric. In particular, every embedded smooth submanifold 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.

Isometries

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

gM=f∗gN,{displaystyle g^{M}=f^{*}g^{N},,}

or pointwise

gpM(u,v)=gf(p)N(df(u),df(v))∀p∈M,∀u,v∈TpM.{displaystyle g_{p}^{M}(u,v)=g_{f(p)}^{N}(df(u),df(v))qquad forall pin M,forall u,vin T_{p}M.}

Moreover, a differentiable mapping f : M → N is called a local isometry at p ∈ M if there is a neighbourhood U ⊂ M, p ∈ U, such that f : U → 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 arc length of a minimizing geodesic. Moreover, this metric space's natural topology agrees with the manifold's topology.^[1]

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

Lab(c):=∫abg(c′(t),c′(t))dt=∫ab‖c′(t)‖dt.{displaystyle 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 particular, a curve [a, b] → M can be parametrized by its arc length. A curve is parametrized by arclength if and only if ‖c′(t)‖ = 1 for all t ∈ [a, b].

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

d(p,q)=infL(γ){displaystyle d(p,q)=inf L(gamma )}

where the 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, 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

diam(M):=supp,q∈Md(p,q)∈R≥0∪{+∞}.{displaystyle mathrm {diam} (M):=sup _{p,qin M}d(p,q)in mathbf {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.

Geodesic completeness

A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map exp_p is defined for all v ∈ T_pM, 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 which are not complete.

See also

• Riemannian geometry


• Finsler manifold


• Sub-Riemannian manifold


• Pseudo-Riemannian manifold


• Metric tensor


• Hermitian manifold


• Space (mathematics)


• Wave maps equation

References

• 
  Jost, Jürgen (2008), Riemannian Geometry and Geometric Analysis (5th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-77340-5.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  do Carmo, Manfredo (1992), Riemannian geometry, Basel, Boston, Berlin: Birkhäuser, ISBN 978-0-8176-3490-2
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External links

• 
  L.A. Sidorov (2001) [1994], "Riemannian metric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4