Riemannian manifold

A Riemannian manifold or a Riemannian space is an object from the mathematical sub-area of Riemannian geometry . These manifolds have the additional property that they have a metric similar to a Prehilbert space . The essential geometric properties of the manifold can then be described with the help of this Riemannian metric . Thus the following, partly equivalent, properties hold for every Riemannian manifold:

• The shortest routes between different points (the so-called geodesics ) are not necessarily straight lines, but can be curved curves.
• In contrast to the plane, the sum of the angles of triangles can also be larger (e.g. sphere) or smaller (hyperbolic spaces) than 180 °.
• The parallel translation of tangent vectors along closed curves can change the direction of the vector.
• The result of a parallel shift of a tangential vector also depends on the path along which the tangential vector is shifted.
• The curvature is generally a function of the location on the manifold.
• Distance measurements between different points are only possible with the help of a metric that can depend on the location on the manifold.

The somewhat more general concept of the pseudo-Riemannian or semi-Riemannian manifold is of decisive importance in the general theory of relativity , since in this the space-time is described as such.

definition

A Riemannian manifold is a differentiable -dimensional manifold with a function that assigns a scalar product of the tangent space to each point , i.e. a positively definite , symmetrical bilinear form${\ displaystyle n}$ ${\ displaystyle M}$${\ displaystyle g}$${\ displaystyle p \ in M}$ ${\ displaystyle T_ {p} M}$

${\ displaystyle g_ {p} \ colon T_ {p} M \ times T_ {p} M \ to \ mathbb {R}}$,

which depends differentiable on. That is, given the differentiable vector fields is ${\ displaystyle p}$ ${\ displaystyle X, \, Y \ in {\ mathfrak {X}} (M)}$

{\ displaystyle {\ begin {aligned} M & \ to \ mathbb {R} \\ p & \ mapsto g_ {p} (X_ {p}, Y_ {p}) \ end {aligned}}}

a differentiable function. The function is called a Riemannian metric or metric tensor , but is not a metric in the sense of the metric spaces . ${\ displaystyle g}$

Examples

Euclidean vector space

A Euclidean vector space is isometrically isomorphic to the standard scalar product${\ displaystyle \ mathbb {R} ^ {n}}$

${\ displaystyle \ langle (x_ {1}, \ dotsc, x_ {n}), (y_ {1}, \ dotsc, y_ {n}) \ rangle = x_ {1} y_ {1} + \ dotsb + x_ {n} y_ {n}}$.

The vector space can be understood as a differentiable manifold and together with the standard scalar product it becomes a Riemannian manifold. In this case the tangential space is identical to the starting space , i.e. again the . ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle T_ {x} \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {R} ^ {n}}$

Induced metric

Since the tangent bundle of a submanifold of a Riemannian manifold is also a subset of the tangential bundle of , the metric of can also be applied to the tangent vectors of the submanifold . The metric of the submanifold obtained in this way is therefore also called the induced metric. The submanifold together with the induced metric again forms a Riemannian manifold. ${\ displaystyle TN}$${\ displaystyle M}$${\ displaystyle TM}$${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle N}$${\ displaystyle N}$

Induced metrics are used as a submanifold${\ displaystyle \ mathbb {R} ^ {n}}$ in the geometric investigation of curves and surfaces .

Riemannian manifolds as metric spaces

The Riemannian metric is not a metric in the sense of the theory of metric spaces, but a scalar product . However, similar to the theory of the scalar product spaces, one can obtain a metric from the scalar product. Thus, Riemannian manifolds can be understood as metric spaces . In contrast to differentiable manifolds, concepts such as distance, diameter or completeness are defined on Riemannian manifolds.

Distance function

Let the following be a Riemannian manifold. The distance function on a ( connected ) Riemannian manifold is then defined by ${\ displaystyle (M, g)}$

${\ displaystyle d (x, y): = \ inf \ {L (\ gamma) \ mid \ gamma \ colon [0,1] \ to M, \ gamma (0) = x, \ gamma (1) = y \}}$.

It runs through all (piece-wise) differentiable paths that connect and , and denotes the length of , according to ${\ displaystyle \ gamma}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle L (\ gamma)}$${\ displaystyle \ gamma}$

${\ displaystyle L (\ gamma) = \ int _ {0} ^ {1} \! {\ sqrt {g _ {\ gamma (t)} ({\ dot {\ gamma}} (t), {\ dot { \ gamma}} (t))}} \, \ mathrm {d} t}$

is defined. The functional is also called length functional. A path that is traversed at constant speed and realizes the shortest connection locally (i.e. for points that are sufficiently close together) is called a geodetic . ${\ displaystyle L}$

The metric thus defined induces the original topology of . Since it can be shown that every differentiable -dimensional manifold has Riemannian metrics, it can also be shown that every differentiable -dimensional manifold is metrizable . Similar to metric vector spaces, one can also speak of complete Riemannian manifolds. The Hopf-Rinow theorem is the central result regarding the completeness of Riemannian manifolds. ${\ displaystyle d}$${\ displaystyle M}$${\ displaystyle n}$${\ displaystyle n}$

diameter

Just like in the theory of metric spaces, is made by

${\ displaystyle \ operatorname {diam} (M): = \ sup \ {d (p, q) \ mid p, q \ in M ​​\} \ in \ mathbb {R} _ {\ geq 0} \ cup \ { \ infty \}}$

defines the diameter of a Riemannian manifold . ${\ displaystyle (M, g)}$

The diameter is an invariant of a Riemannian manifold under global isometries. In addition, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds , that is, a complete Riemannian manifold is compact if and only if the diameter is finite.

history

Gauss' theory of curved surfaces uses an extrinsic description, that is, the curved surfaces are described with the help of a surrounding Euclidean space. Riemann, on the other hand, takes a more abstract approach. Riemann introduced this approach and the associated definitions in his habilitation lecture on the hypotheses on which geometry is based on June 10, 1854 at the University of Göttingen . Many definitions were also presented there that are still used in modern mathematics today. Of paracompact spaces at that time had not yet spoken. Instead of curves and tangential vectors, Riemann used infinitesimal line elements.

So - called non - Euclidean geometries have been discussed since the beginning of the 19th century . The Riemannian geometry has just the right definitions and the right language to describe these geometries from a general point of view. The concept of the Riemannian manifold formed a fundamental starting point for the development of the general theory of relativity at the beginning of the 20th century .

literature

• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8
• Marcel Berger : A panoramic view of Riemannian geometry . Springer-Verlag, Berlin, 2003, ISBN 3-540-65317-1
• Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine: Riemannian Geometry (Second Edition), Springer-Verlag, Berlin / Heidelberg 1990, ISBN 3-540-52401-0
• Martin Schottenloher: Geometry and Symmetry in Physics , vieweg textbook, 1995, ISBN 3-528-06565-6