This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a Riemannian Connection on the ambient manifold $\bar M$, $D$ being the Riemannian Connection on M and V being a symmetric vector values $(0,2)\;C^\infty$ tensor. I dont understand how that decomposition into tangential and normal components is brought about??I know that every vector can be written as sum of tangential and normal components but not sure how to relate that here.
Now in the proof , we employ Gram-Schimdt orthogonalisation process to obtain orthonormal vector fields $ W_1,\ldots,W_n$ on $\bar U$ such that $W_1|_U,\ldots,W_k|_U$ are orthonormal base for $M_m ,m\in U$, while $W_{k+1}|_U,\ldots,W_{n}|_U$ give $\bar M$-vector fields on $U$ which give a base for the orthogonal complement to $M_m$.I do not understand the last part of this argument with regards to the complement.Finally I let $\bar D_{W_i}W_j=\sum_{r=1}^{n}B_{ij}^{r}$ on $\bar U$. Then $$\bar D_X Y =\sum(XY_j)W_j+\sum Y_jB_{ji}^{r}W_r$$
Can someone explain what the $B_{ij}^{r}$ are , though they look like Christoffel symbols to me. The book concludes this as the result.But I am unable to do the same. Can anyone please help?