**Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866)**was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.

__Euclidean geometry versus Riemannian geometry__In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture

at Göttingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important

works in geometry. It was titled Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely:

"On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.

The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.

__Higher dimensions__Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.

__Riemannian metrics__Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive definite) inner products

**[g_{p}] : [T_{p}][M] x [T_{p}][M] --> R, p in M**

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

**p --> [g_{p}](X(p), Y(p))**

defines a

**--> R, R being the set of real numbers.**

__smooth function M__More formally:

**A Riemannian metric g is a section of the vector bundle [S^(2)][T*][M].**

In a system of local coordinates on the manifold M given by n real-valued functions

**x^(1), x^(2), ..., x^(n)**, the vector fields

**{ Partial derivative w.r.t. x^(1), ...,**

**Partial derivative w.r.t. x^(n) }**

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,

**[g_{(i, j)}] := [g_p] [ ( Partial derivative w.r.t. x^(i))_{p}, (Partial derivative w.r.t. x^(j))_{p} ) ].**

Equivalently, the metric tensor can be written in terms of the dual basis {dx^(1), ..., dx^(n)} of the cotangent bundle as

**g = Summation_{for all (i, j)}**

**{ ( [g_{(i, j)}] dx^(i) (x) dx(j) ) }**

Endowed with this metric, the differentiable manifold (M, g) is a Riemannian manifold.

Let (M, g) be a Riemannian manifold and N (a subset of 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 : M^(n) --> N^(n + k) be an immersion. Then, if N has a Riemannian metric, f induces a Riemannian metric on M via pullback:

**[g_{p}]^(M) : [T_{p}][M] x [T_{p}][M] --> R**

**(R is the set of real numbers)**

**(u, v) --> {[g_{p}]^(M)}(u, v)**

**:= [g_{f(p)}]^(N) ([T_{p}]f(u),[T_{p}]f(v)).**

This is then a metric; its positive definiteness follows from the injectivity

of the differential of an immersion.

Let (M, g^(M)) be a Riemannian manifold, h: M^(n + k) --> N^(k)

a differentiable map and q (in N) a regular value of h (the differential

dh(p) is surjective for all p in h^(-1)(q)). Then {[h^(-1)](q)} (which is a

subset of M) is a submanifold of M of dimension n. Thus [h^(-1)](q)

carries the Riemannian metric induced by inclusion.

__The pullback metric__If f : M --> N is a differentiable map and (N, g^(N)) is a Riemannian

manifold, then the pullback of g^(N) along f is a

**quadratic form**

**on the**

**tangent space TM of M**. The

**pullback**is the quadratic

form f*g^(N) on TM defined over elements v, w of [T_{p}][M]

by

**( 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.**

To prove this result, let M be a manifold and

**{(U_{a}, PHI[U_{a}]) | a is in I}**

a

**locally finite atlas of open subsets U of M**and diffeomorphisms

onto open subsets of R^(n)

**PHI := U_a --> PHI(U_a) (which is a subset of R^(n)).**

Let

**[t_{a}]**be a

__differentiable partition of unity subordinate to__**. Then define the metric g on M by**

__the given atlas__**g := Summation_{beta} [(t_{beta})(f_{beta})],**

**with (f_{beta})**

**:= {slur[PHI*]_{beta}}(g^(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 --> N be a diffeomorphism. Then, f is called an isometry, if

**g^(M) = f*g^(N)**,

or pointwise

**{[g_{p}]^(M)}(u, v) = {[g_{f(p)}]^(N)}([T_{p}](f(u)),[T_{p}](f(v))**

**for ALL p in M, AND for ALL u and v in [T_{p}](M).**

Moreover, a differentiable mapping f : M --> N is called a

**local isometry at p**(element of M) if there is a neighborhood U

(which is a subset of M), with U containing p 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 arclength of a minimizing

geodesic. 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

__is defined__

**length of c (L(C)**as

**[L_{a}^{b}](c) := Integral_{a}^{b} {SQRT[g(c'(t),c'(t))] dt**

**= Integral_{a}^{b} {(c'(t)) dt}**

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 its arclength L(C) if and only if**

**c'(t) = 1 for**

**all t in [a, b].**

The distance function d : M x M --> [0, infinity) is defined by

**d(p, q) = inf L(C)**

where the infimum extends over all differentiable curves C beginning

at p (in M) and ending at q (also in 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) := sup_{p, q elements of M} {d(p, q)}**

**(diam(M) is an element of [R_{>= 0}] Union {+infinity}).**

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 in M,

the exponential map (exp_{p}) is defined for all v in [T_{p}][M],

i.e. if any geodesic C(t) starting from p is defined for all values of the

real-valued parameter t.

**The**

**Hopf-Rinow theorem**

**asserts that M is geodesically complete**

**if and only if it is**

**complete as a metric space.**

This theorem is named after Heinz Hopf and his student Willi Rinow

(1907–1979).

__Statement of the Hopf-Rinow theorem__**Let (M, g) be a finite-dimensional Riemannian manifold. Then the**

**following statements are equivalent:**

**1. The closed and bounded subsets of M are compact;**

**2. M is a complete metric space;**

**3. M is geodesically complete; that is, for every p in M, the**

**exponential map exp(p) is**

**defined on the**

**entire tangent space**

**[T_{p}](M).**

**Furthermore, any one of the above implies that given any two**

**points**

**p and q in M, there exists a length minimizing geodesic**

**connecting**

**these two points (geodesics are in general extrema,**

**and may or may**

**not be minima). This does not hold in infinite**

**dimensions:**

**(Atkin 1975)**

**showed that two points in an infinite**

**dimensional complete Riemannian**

**manifold need not be connected**

**by any geodesic.**

__Generalization of the HOPF-RINOW THEOREM__The Hopf–Rinow theorem is generalized to length-metric spaces the

following way:

**If a length-metric space (M, d) is complete and locally compact**

**then any**

**two points in M can be connected by minimizing geodesic,**

**and any bounded**

**closed set in M is compact.**

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.**

Source: Wikipedia article

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