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4.11.10

The Mathematics of Georg Friedrich Bernhard Riemann

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 smooth function M --> R,  R being the set of real numbers.

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
the given atlas.   Then define the metric g on M by

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 length of c (L(C) is defined
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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