Tangential component of a Riemannian connection Let $f: M \to N$ be an immersion of a differentiable manifold $M$ into a Riemannian manifold $N$. Assume that $M$ has the
Riemannian metric induced by $f$.
Let $p \in M$ and let $U \subset М$ be a neighborhood of $p$ such that
$f(U) \subset N$ is a submanifold of $M$. Further, suppose that $X, Y$
are differentiable vector fields on $f(U)$ which extend to
differentiable vector fields $X^*, Y^*$ on an open set of $N$. Define
$$(\nabla_x Y)(p) =\text{ tangential component of }(\overline{\nabla}_{x^*} Y^*)(p),$$ where $\overline{\nabla}$ is the
Riemannian connection of $N$. Prove that $\nabla$ is the Riemannian
connection of $M$.
 A: As in the problem statement, we let $\overline{\nabla}$ denote the Riemannian connection on $N$, and define $(\nabla_XY)(p) = \pi^\top[\overline{\nabla}_{X^*}Y^*(p)]$, where $\pi^\top\colon TN|_{f(U)} \to TM$ denotes the tangential projection.  We aim to show that $\nabla$ is the Riemannian connection on $M$.
By the uniqueness of the Riemannian connection on $M$, it suffices to show that (1) $\nabla$ is a connection, (2) $\nabla$ is compatible with the metric, and (3) $\nabla$ is symmetric.
(1) $\nabla$ is a connection
Let $X_1, X_2, Y_1, Y_2$ be differentiable vector fields on $f(U)$ that extend to differentiable vector fields on an open subset of $N$.  We note that
$$(\nabla_{X_1 + X_2}Y)(p) = (\overline{\nabla}_{X_1^* + X_2^*}Y^*)(p) = (\overline{\nabla}_{X_1^*}Y^* + \overline{\nabla}_{X_2^*}Y^*)(p)^\top = (\nabla_{X_1}Y)(p) + (\nabla_{X_2}Y)(p)$$
and
$$(\nabla_X(Y_1 + Y_2))(p) = (\overline{\nabla}_{X^*}Y_1^* + \overline{\nabla}_{X^*}Y_2^*)(p)^\top = (\nabla_XY_1)(p) + (\nabla_XY_2)(p)$$
and if $h\in C^\infty(f(U))$ is any differentiable function on $f(U)$, then
$$\begin{align*}
(\nabla_X(hY))(p) & = (\overline{\nabla}_{X^*}(hY^*))(p)^\top \\
& = \left[(\overline{\nabla}_{X^*}h)(p)Y^*(p) + h(p)(\overline{\nabla}_{X^*}Y^*)(p) \right]^\top \\
& = \left[ (Xh)(p)\,Y^*(p)\right]^\top + h(p)(\overline{\nabla}_{X^*}Y^*)(p)^\top \\
& = (Xh)(p)\,Y(p) + h(p)\,(\nabla_XY)(p),
\end{align*}$$
which shows that $\nabla$ is a connection.
(2) $\nabla$ is compatible with the metric.
This is a computation:
$$\begin{align*}
X_p\langle Y_p, Z_p \rangle_g & = X^*_p\langle Y^*_p, Z^*_p \rangle_{\overline{g}} \\
& = \overline{\nabla}_{X^*}\langle Y^*, Z^* \rangle_{\overline{g}}(p) \\
& = \langle (\overline{\nabla}_{X^*}Y^*)(p), Z^*_p\rangle_{\overline{g}} + \langle Y^*_p, (\overline{\nabla}_{X^*}Z^*)(p)\rangle_{\overline{g}} \\
& = \langle (\overline{\nabla}_{X^*}Y^*)(p)^\top + (\overline{\nabla}_{X^*}Y^*)(p)^\perp, Z_p \rangle_{\overline{g}} + \langle Y_p, (\overline{\nabla}_{X^*}Z^*)(p)^\top + (\overline{\nabla}_{X^*}Z^*)(p)^\perp \rangle_{\overline{g}} \\
& = \langle (\nabla_XY)(p), Z_p \rangle_g  + \langle Y_p, (\nabla_XZ)(p)\rangle_g,
\end{align*}$$
where I've switched to subscripts $X_p = X(p)$ for a slight ease of notation.  (Also, $\perp$ denotes the normal projection.)
(3) $\nabla$ is symmetric.
Finally, symmetry follows from noting that
$$\begin{align*}
(\nabla_XY)(p) - (\nabla_YX)(p) & = (\overline{\nabla}_{X^*}Y^*)(p)^\top - (\overline{\nabla}_{Y^*}X^*)(p)^\top \\
& = \pi^\top\!\left( (\overline{\nabla}_{X^*}Y^*)(p) - (\overline{\nabla}_{Y^*}X^*)(p) \right) \\
& = \pi^\top([X^*, Y^*]_p) \\
& = \pi^\top([X, Y]_p) \\
& = [X, Y]_p
\end{align*}$$
since $[X,Y]$ is tangent to $M$ whenever $X$ and $Y$ are.
