Department of Mathematical Sciences

School of Natural Sciences and Mathematics

Faculty and research

Istvan Ozsvath, Ph.D.

Professor

Education

Ph.D., Hamburg, 1960.

Overview

Relativistic cosmology, differential geometry.

Research Interests

Dr. Ozsvath's research involves two major fields:

1) Exact Solutions of Einstein's Field Equations

Ozsvath constructed a large number of homogeneous solutions, in which the underlying four-manifold is a four-parametric Lie group endowed with a left invariant metric satisfying the field equations.

The enormous significance of these solutions is that they supply examples or counter examples in deciding fundamental questions of Einstein's theory. The two solutions, constructed by I. Ozsvath and E. L. Schucking, illustrate this point.

The first one is the Finite Rotating Universe, which is a perfectly acceptable model of the Universe within Einstein's theory of gravitation. In this model, the Weltsubstrat rotates with respect to the local inertial frame. This is contrary to the "Mach's principle," which requires that a correct gravitational theory should not admit rotating models.

Another of their solutions, the Anti-Mach Metric, called even by that name, is a vacuum solution, free of singularities, geodesically complete, and curved. These facts show that one can have, within Einstein's theory, a genuine gravitational field without sources.

By now, all the homogeneous solutions of the Einstein's field equation have been constructed, most of them by Ozsvath.

The most interesting among the spatially homogeneous models is the one in which the metric on the space-like hypersurfaces is invariant under S3, the 3-sphere. This fact lead Ozsvath to his second field of interest:

2) Embedding Problems of Compact 3-Manifolds

The 3-manifold is the S3 endowed with a left invariant metric. S3 has finite subgroups listed already by Felix Klein. One can use these subgroups to identify points. For example, if one embeds SI, endowed with a specific left invariant metric, into S4 as a hypersurface, one finds that this compact hypersurface has the topology of the Poincare cube. Further calculation leads to yet another embedding in which S8 is the ambient space into which S3, endowed with the most general left invariant metric, is embedded and has the topology of the three dimensional projective space. These results are of great importance if one believes that the Universe is finite.

Future Plans

Ozsvath's most immediate future plans are to cooperate with Engelbert Schucking in studying the SU3 group, due to the importance of this group in high energy physics.

Selected Publications:

  1. Ozsvath, I., and E. Schucking. The world viewed from outside. Submitted.
  2. Ozsvath, I. Embedding of the finite rotating universe. Submitted.
  • Updated: February 6, 2006