Pseudo-Riemannian manifold

A pseudo-Riemannian manifold or semi-Riemannian manifold is a mathematical object from (pseudo) Riemannian geometry . It is a generalization of the Riemannian manifold defined earlier and was introduced by Albert Einstein for his general theory of relativity . However, the object was named after Bernhard Riemann , the founder of Riemannian geometry. But a structure of a manifold was also named after Albert Einstein. These Einsteinian manifolds are a special case of the pseudo-Riemannian.

definition

In the following, the tangential space at a point of a differentiable manifold is denoted by. A pseudo-Riemannian manifold is a differentiable manifold together with a function . This function is tensorial , balanced and not degenerated , ie for all tangent vectors and functions applies ${\ displaystyle T_ {p} M}$ ${\ displaystyle p}$ ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle g_ {p} \ colon T_ {p} M \ times T_ {p} M \ to \ mathbb {R}}$ ${\ displaystyle X, \ Y, \ Z \ in T_ {p} M}$ ${\ displaystyle a, \ b \ in C ^ {\ infty} (M)}$

${\ displaystyle g_ {p} (aX + bY, Z) = a (p) \, g_ {p} (X, Z) + b (p) \, g_ {p} (Y, Z)}$ (tensorial),
${\ displaystyle g_ {p} (X, Y) = g_ {p} (Y, X)}$ (symmetrical),
if for is that for all , then it follows . ${\ displaystyle X \ in T_ {p} M}$ ${\ displaystyle g_ {p} (X, Y) = 0}$ ${\ displaystyle Y}$ ${\ displaystyle X = 0}$

In addition, is differentiable dependent on . The function is thus a differentiable tensor field and is called pseudo-Riemannian metric or metric tensor . ${\ displaystyle g_ {p}}$ ${\ displaystyle p}$ ${\ displaystyle g}$ ${\ displaystyle g \ colon TM \ times TM \ to C ^ {\ infty} (M)}$

signature

As with any common bilinear form , the pseudo-Riemannian metric can also be assigned a signature . Due to Sylvester's law of inertia, this is independent of the choice of the coordinate system on the manifold and thus also independent of the choice of the point . As with the determinant, there are numerous equivalent terms for a given “physics”. But since it has not degenerated, the third entry in the signature is always zero and the determinant of is always not equal to zero. Four-dimensional pseudo-Riemannian manifolds with the signature (3,1,0) (or mostly (1,3,0)) are called Lorentz manifolds . These play an important role in general relativity. ${\ displaystyle p \ in M}$ ${\ displaystyle g}$ ${\ displaystyle g}$

Pseudo-Riemannian geometry

In contrast to pseudo-Riemannian metrics, the Riemannian metrics are positive and definite , which is a stronger requirement than “not degenerated”. Some results from Riemannian geometry can also be transferred to pseudo-Riemannian manifolds. For example, the main theorem of Riemannian geometry also applies to pseudo-Riemannian manifolds. There is therefore a clear Levi-Civita connection for every pseudo-Riemannian manifold . However, in contrast to Riemannian geometry, one cannot find a metric with a given signature for every differentiable manifold. Another important difference between Riemannian and pseudo-Riemannian geometry is the lack of an equivalent for the Hopf-Rinow theorem in pseudo-Riemannian geometry. In general, metric completeness and geodetic completeness are not linked here. The signature of the metric also gives rise to problems for the continuity of the distance function. For example, the distance function for Lorentz manifolds can have the property of not being above semi-continuous .

Definition variant

Deviating from the above definition, Serge Lang distinguishes between semi-Riemannian and pseudo-Riemannian manifolds and additionally requires that the former be positive semidefinite , i.e. for all . ${\ displaystyle g_ {p}}$ ${\ displaystyle g_ {p} (X, X) \ geq 0}$ ${\ displaystyle X \ in T_ {p} M}$

Individual evidence

^ Serge Lang: Differential and Riemannian manifolds. 3. Edition. Springer Science + Business Media, New York 1995, ISBN 0-387-94338-2 , p. 30 ( Graduate Texts in Mathematics. Volume 160, limited preview in Google Book Search)

literature

Manfredo Perdigão do Carmo: Riemannian Geometry ("Geometria Riemannia"). 2nd ed. Birkhäuser, Boston 1993, ISBN 0-8176-3490-8 .
Peter Petersen: Riemannian geometry (Graduate Texts in Mathematics; Vol. 171). 2nd edition Springer-Verlag, New York 2006, ISBN 0-387-29403-1 .
John K. Beem, Paul E. Ehrlich, Kevin L. Easley: Global Lorentzian Geometry (Pure and Applied Mathematics; Vol. 202). 2nd edition Marcel Dekker Books, New York 1996, ISBN 0-8247-9324-2 .

[1] Serge Lang: Differential and Riemannian manifolds. 3. Edition. Springer Science + Business Media, New York 1995, ISBN 0-387-94338-2 , p. 30 ( Graduate Texts in Mathematics. Volume 160, limited preview in Google Book Search)