Tangential component of vector field

Question

I'd like to know the definition of "tangential component" in this case, it is question 3 of page 57 of Do Carmo's book Riemannian Geometry:

It says:

Define $\nabla_XY(p) = $ tangential component of $\overline\nabla_{\overline X}\overline Y(p) $, where $\overline\nabla$ is the Riemannian connection of $\overline M$.

Answer 1

You have a submanifold $M$ of a bigger manifold $\overline{M}$. Tangential means "tangent to $M$".

Specifically, at each point $p\in M$, you have a map $\pi:T_p\overline{M}\rightarrow T_p M$ defined as follows.

The metric $g$ gives us an orthogonal complement $\nu$ to $T_ p M$ so that $T_p \overline{M} = T_p M\oplus \nu$. Then $\pi(X_{T_p M} + X_{\nu}) = X_{T_p M}$. The tangential componenet of $X$ is then, by defintion, $\pi(X)$.

Answer 2

The problem is about an immersed submanifold $M^n$ of a Riemannian manifold $\overline{M}^{n+k}$. This immersion induces an injection $$T_*f: T_p M \rightarrow T_{f(p)}\overline{M}$$ ($f$ denoting the injection) which allows you to identify the tangent space of $M$ with a subspace of $T\overline{M}$. If now $q = f(p)\in T_{q}\overline{M}$ and $\overline{Z}\in T_{q}\overline{M}$ the tangential component of $\overline{Z}$ is just the orthogonal projection of $\overline{Z}$ with respect to the metric on $T_q\overline{M}$ onto the tangent space of (the image of) $M$.