I recently saw for the first time formulas that are covariant versions of Taylor series. Because they are not easy to find on the internet, I write some down here. Suppose $x_0$ and $x_1$ are two points on a manifold, then the covariant Taylor series are formulas like
\begin{align*}
f(x_1) &= f(x_0) + f_{;\mu}(x_0)\, \eta^{\mu} + \dfrac{1}{2}f_{;\mu\nu}(x_0)\, \eta^{\mu}\eta^{\nu} + O(\eta)^3\\
T_{\mu}(x_1) &= T_{\mu}(x_0) + T_{\mu;\alpha}(x_0)\, \eta^{\alpha} +
\dfrac{1}{2}\left( T_{\mu;\alpha\beta}(x_0)+\dfrac{1}{3} R^{\sigma}_{\ \ \alpha\beta\mu}(x_0)\, T_{\sigma}(x_0)\right) \eta^{\alpha}\eta^{\beta} + O(\eta)^3
\end{align*}
The semi-colon denotes the covariant derivative with the Levi-Civita connection. The vector $\eta^{\mu}$ is defined as follows: take a geodesic $\gamma(t)$ such that $\gamma(0) = x_0$ and $\gamma(1) = x_1$, then $\eta^{\mu} = \dot\gamma^{\mu}(0)$. The higher coefficients in the series expansion become more and more complicated formulas involving the Riemann tensor and its covariant derivatives. The formulas can be proved using normal coordinates.

More information can be found in "The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model", by
Alvarez-Gaume, Freedman, Mukhi, 1981

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