Distorted product

In mathematics and physics, especially in differential geometry and general relativity , which referred distorted product of two pseudo-Riemannian manifolds the product diversity with the distorted product metric .

definition

By the distorted product of two pseudo-Riemannian manifolds and along a strictly positive function one understands the product manifold equipped with the metric tensor . And denote the natural submersions and the pullback of a tensor under a mapping g between two manifolds. In this case, as a base and as a fiber denotes the product manifold. ${\displaystyle M\times _{f}N}$ ${\displaystyle (M,g_{M})}$ ${\displaystyle (N,g_{N})}$ ${\displaystyle f\colon M\to (0;\infty )}$ ${\displaystyle M\times N}$ ${\displaystyle g:=\pi ^{*}(g_{M})+(f\circ \pi )^{2}\sigma ^{*}(g_{N})}$ ${\displaystyle \pi \colon M\times N\to M}$ ${\displaystyle \sigma \colon M\times N\to N}$ ${\displaystyle g^{*}}$ ${\displaystyle M}$ ${\displaystyle N}$

Definition of skewed metric

A distorted product metric is understood to be a Riemann or Lorentz manifold , the metric of which is through

${\displaystyle ds^{2}\,=g_{ab}(y)dy^{a}dy^{b}+f(y)g_{ij}(x)dx^{i}dx^{j}}$

can be represented. I.e. in particular, the manifold under consideration breaks down into the Cartesian product of a “y” and an “x” geometry, whereby the “x” metric is distorted.

literature

• Barrett O'Neill: Semi-Riemannian Geometry. With Applications to Relativity (Pure and applied mathematics; Vol. 103). Academic Press, New York 1983, ISBN 0-12-526740-1 .