The Taylor series related to the differential of the exponential map

TL;DR

This paper derives an explicit Taylor series expansion for an operator related to the differential of the exponential map on a manifold, expressed through curvature tensors and their derivatives.

Contribution

It provides a new explicit formula for the Taylor series of the differential of the exponential map involving curvature and its derivatives.

Findings

Explicit Taylor series formula derived

Expressed in terms of curvature tensor and derivatives

Enhances understanding of exponential map behavior

Abstract

In this paper we study the Taylor series of an operator-valued function related to the differential of the exponential map. For a smooth manifold $\mathcal{M}$ with a torsion-free affine connection the operator $\mathcal{E}_p(v)$ acting on the space $T_p\mathcal{M}$ is defined to be the composition of the differential of the exponential map at $v\in T_p\mathcal{M}$ with parallel transport to $p$ along the geodesic. The Taylor series of $\mathcal{E}_p$ as a function of $v$ is found explicitly in terms of the curvature tensor and its high order covariant derivatives at $p$.