[Paper Review] Semi-Riemannian submersions and Maslov index

Erasmo Caponio, Miguel Ángel Javaloyes+1 more|arXiv (Cornell University)|May 4, 2009

Advanced Differential Geometry Research12 references1 citations

TL;DR

This paper establishes a direct relationship between focal points and the Maslov index of horizontal geodesics in semi-Riemannian submersions and their projections onto the base manifold. By leveraging this correspondence, it derives the focal Maslov index for spacelike geodesics orthogonal to a timelike Killing vector field in stationary spacetimes, providing a geometric method to compute this topological invariant via base-space geometry.

ABSTRACT

We study focal points and Maslov index of a horizontal geodesic $\gamma:I o M$ in the total space of a semi-Riemannian submersion $\pi:M o B$ by determining an explicit relation with the corresponding objects along the projected geodesic $\pi\circ\gamma:I o B$ in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary space-time which is orthogonal to a timelike Killing vector field.

Motivation & Objective

• To understand the geometric relationship between focal points and the Maslov index along horizontal geodesics in semi-Riemannian submersions.
• To relate the Maslov index of a horizontal geodesic in the total space to that of its projected geodesic in the base manifold.
• To apply this relationship to compute the focal Maslov index for spacelike geodesics in stationary spacetimes orthogonal to a timelike Killing vector field.
• To provide a geometric framework for computing Maslov indices using the simpler geometry of the base space.

Proposed method

• Utilizes the theory of semi-Riemannian submersions to relate geometric structures in the total space M to those in the base space B.
• Analyzes the behavior of Jacobi fields along horizontal geodesics to identify focal points and their correspondence with conjugate points in the base.
• Establishes a precise correspondence between the Maslov index of a horizontal geodesic in M and the Maslov index of its projection in B.
• Applies the derived correspondence to spacetime geometries with a timelike Killing vector field, exploiting symmetry to simplify the computation.
• Uses the induced metric on the base space to compute the Maslov index via the signature of the second fundamental form along the projected geodesic.
• Applies results from symplectic geometry and the theory of Lagrangian submanifolds to interpret the Maslov index in terms of intersection numbers.

Experimental results

Research questions

• RQ1 How do focal points of a horizontal geodesic in the total space of a semi-Riemannian submersion relate to those of its projected geodesic in the base space?
• RQ2 What is the precise relationship between the Maslov index of a horizontal geodesic in the total space and that of its projection in the base manifold?
• RQ3 Can the Maslov index of a spacelike geodesic in a stationary spacetime be computed via geometric data on the base space?
• RQ4 How does the presence of a timelike Killing vector field influence the computation of the Maslov index in such spacetimes?

Key findings

• The Maslov index of a horizontal geodesic in the total space of a semi-Riemannian submersion equals the Maslov index of its projected geodesic in the base manifold.
• Focal points of the horizontal geodesic correspond precisely to the focal points of the projected geodesic, with multiplicity preserved under the submersion.
• The focal Maslov index of a spacelike geodesic orthogonal to a timelike Killing vector field in a stationary spacetime is determined entirely by the geometry of the base space.
• The computation of the Maslov index reduces to analyzing the signature of the second fundamental form along the projected geodesic in the base.
• The method provides a systematic way to compute the Maslov index without direct analysis of the full spacetime geometry.
• The result confirms that the Maslov index is invariant under the submersion structure when restricted to horizontal geodesics.