Inequalities for Relativistic Embeddings in non-compact Kaluza-Klein Theories

In recent years there has been a growing interest in the study of space-times embedded in some higher-dimensional space (see e.g. brane models, induced matter theory, ...). In induced matter theory for example, a space-time is considered to be a hypersurface in a five-dimensional space with vanishing Ricci tensor, such that the fifth variable, i.e. a matter coordinate, is constant on the hypersurface. The theory however provides no way to obtain a unique space-time from a given five-dimensional metric. At this stage one could look for mathematical conditions on the embedding to limit the possible space-times.

We present two such conditions. First we obtain inequalities between intrinsic and extrinsic scalar curvature invariants related to the embeddings and show that the equality holds if and only if the second fundamental form of the embeddings has a specific form. A first inequality holds for space-times in some higher-dimensional space such that the normal directions have the same signature and it relates a new intrinsic scalar curvature invariant to the squared mean curvature. Space-times which satisfy the equality are called ideal because the tension they receive from the surrounding space is minimal. The second inequality holds for space-time embedded in a space with codimension two and relates the intrinsic scalar curvature of the space-time to the extrinsic squared mean curvature and the scalar normal curvature. In both cases we give examples of space-times which satisfy the equality.

 

Fecha: 
23/06/2005 - 14:00
Conferenciante: 
Stefan Haesen
Filiación: 
Dept. of Mathematics, Katholieke Universiteit Leuven


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