Uniform Parametrization in Pseudo-Complex Hyperbolic Space

TL;DR

This paper introduces a uniform parametrization theorem for pseudo-complex hyperbolic spaces with multiple time-like dimensions, preserving inner products and invariance under linear transformations, expanding the mathematical framework of such spaces.

Contribution

It derives a new parametrization theorem for pseudo-complex hyperbolic spaces with multiple time-like dimensions, ensuring invariance and preserving inner products.

Findings

Uniform parametrization for time-like and space-like dimensions.

Invariance of space elements under linear transformations.

Preservation of complex-hyperbolic inner product.

Abstract

The parametrization theorem is derived in a flat nD pseudo-complex affine space. The pseudo-complex hyperbolic space accomodates n-number of uncompactified time-like extra dimensions with sugnature (s,r), where s and r are the numbers of minus and plus signs associated with the diagonalized metric matrix. The main result of the theorem suggests a uniform parametrization for both time-like and space-like dimensions. The uniformization requirement preserves complex-hyperbolic inner product associated with the space. As application, the elements of the space is shown to be invariant under linear transformation.