Geodesic flows on spheres and the local Riemann-Roch numbers

TL;DR

This paper computes local Riemann-Roch numbers for zero sections of cotangent bundles over spheres and real projective spaces, utilizing the $S^1$-bundle structure derived from geodesic flows.

Contribution

It introduces a method to calculate local Riemann-Roch numbers using the $S^1$-bundle structure related to geodesic flows on specific manifolds.

Findings

Calculated local Riemann-Roch numbers for $T^*S^n$ and $T^* P^n$

Established a link between geodesic flows and Riemann-Roch invariants

Provided explicit formulas for these local invariants

Abstract

We calculate the local Riemann-Roch numbers of the zero sections of $T^*S^n$ and $T^*\R P^n$, where the local Riemann-Roch numbers are defined by using the $S^1$-bundle structure on their complements associated to the geodesic flows.