Geodesic flows on spheres and the local Riemann-Roch numbers

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