A formula for the k-th covariant derivative

TL;DR

This paper derives a formula for the k-th covariant derivative of tensor fields along a curve, introducing new symbols based on Christoffel symbols and generalizing the result for real k.

Contribution

It provides a new explicit formula for the k-th covariant derivative along curves, extending previous results to real values of k.

Findings

Introduces symbols P and Q based on Christoffel symbols

Provides a formula for the k-th covariant derivative

Generalizes the formula for real k

Abstract

The aim of the present paper is to give a formula for the $k$-th covariant derivative of tensor field along a given curve. In order to do that, first the symbols $P^{i<k>}_{j}$ and $Q^{i<k>}_{j}$ which depend on the Christoffel symbols are introduced. Some properties of them are also given. The main result is given by (3.1) and further it is generalized for $k\in R$.