Real Projective Geodesics Embedded In Complex Manifolds

TL;DR

This paper investigates the mathematical foundations of modeling space-time with complex manifolds, deriving projective geodesics that include additional effects beyond gravity by projecting complex geodesics onto real sub-manifolds.

Contribution

It derives equations of motion as projective geodesics on real sub-manifolds embedded in complex manifolds, revealing additional terms beyond standard Christoffel symbols.

Findings

Derived complex manifold geodesics using local coordinates.

Projected geodesics onto real sub-manifolds showing extra terms.

Expanded understanding of effects beyond gravity in space-time models.

Abstract

Focus of this study is to explore some aspects of mathematical foundations for using complex manifolds as a model for space-time. More specifically, certain equations of motions have been derived as a Projective geodesic on a real manifold embedded within a complex one. To that goal, first the geodesic on complex manifold has been computed using local complex and conjugate coordinates, and then its projection on the real sub-manifold has been studied. The projective geodesic, thus obtained, is shown to have additional terms beyond the usual Christoffel symbols, and hence expands the geodesic to capture effects beyond the mere gravitational ones.