I'm studying some Riemannian geometry and am having difficulty understanding some calculations and notions with the covariant derivative. In particular I am currently stuck on the following:
Let $S^n \subset \mathbb{R}^{n+1}$ be the usual unit sphere with the induced Riemannian metric from the Euclidean metric of $\mathbb{R}^{n+1}$. If $\gamma : [a,b] \rightarrow S^n$ a curve and $V(t)$ a vector field on $S^n$ along the curve $\gamma$, that is, $V(t) \in T_{\gamma(t)} S^n$ for all $t \in [a,b]$, show the covariant derivative (defined using the Levi-Civita connection on $S^n$ is given by: $$ \frac{DV}{dt} = \text{pr}_{T_{\gamma(t)}S^n}(V'(t)) $$ where pr denotes orthogonal projection and $V'(t)$ is the usual derivative in $\mathbb{R}^{n+1}$. Bonus: generalise this to arbitrary submanifold $N$ of a Riemannian manifold $(M,g)$ with the induced metric on $N$.
My idea was to expand the covariant derivative as follows: $$ \nabla_{\dot{\gamma}} V = \sum_{i=1}^{n} \left(\frac{dV_i}{dt} + \sum_{j,k=1}^{n} \Gamma_{jk}^{i} \frac{d\gamma_j}{dt} V_k(t)\right)\frac{\partial}{\partial x_i} $$ and then hope to simplify this via calculating the Christoffel symbols or something of the kind but I'm having some trouble from here.