Title: Localization and Stationary Phase Approximation on Supermanifolds,
Abstract: Given an odd vector field $Q$ on a supermanifold $M$ and a $Q$-invariant
density $\mu$ on $M$, under certain compactness conditions on $Q$, the value of
the integral $\int_{M}\mu$ is determined by the value of $\mu$ on any
neighborhood of the vanishing locus $N$ of $Q$. We present a formula for the
integral in the case where $N$ is a subsupermanifold which is appropriately
non-degenerate with respect to $Q$.
In the process, we discuss the linear algebra necessary to express our result
in a coordinate independent way. We also extend stationary phase approximation
and the Morse-Bott Lemma to supermanifolds.