Let $(\tilde M, \tilde g)$ be a Riemannian manifold and $M$ be an embedded submanifold of $M$ with the induced Riemannian metric $g$.
On Pg. 138 of Lee's Riemannian Geometry - An Introduction to Curvature, the following is written:
For any vector $V\in T_pM$, $II(V, V)$ is the $\tilde g$-acceleration at $p$ of the $g$-geodesic $\gamma_V$.
Here $II$ denotes the second fundamental form.
This seems to suggest that for vector fields $X$ and $Y$ on $M$, the second fundamental form $II(X, Y)$ at the point $p\in M$ depends only on the value $X(p)$ and $Y(p)$.
But why is this so? By definition $II(X, Y)(p)$ is the orthogonal projection on $(T_pM)^\perp$ of the vector $(\tilde\nabla_X(Y))(p)$. (Here $\tilde\nabla$ is the Riemannian connection on $\tilde M$.)
I know that $(\tilde\nabla_XY)(p)$ depends only on the value taken by $X$ at the point $p$, and the values taken by $Y$ on a curve in $M$ which is tangent to $X(p)$. But $(\tilde\nabla_XY)(p)$ is not determined solely by $X(p)$ and $Y(p)$.
Can somebody please clear this confusion. Thanks.
$\newcommand{\Del}{\tilde{\nabla}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$Let $p$ be a point of $M$, and let $N$ be an arbitrary smooth normal field to $M$ in some neighborhood of $p$. If $Y$ is a tangent field near $p$, then since $\Brak{N, Y} = 0$ pointwise, $$ 0 = X\Brak{Y, N} = \Brak{\Del_{X}Y, N} + \Brak{Y, \Del_{X}N}. $$ Particularly, $\Brak{\Del_{X}Y, N} = -\Brak{Y, \Del_{X}N}$ depends only on the value of $Y$ at $p$. Since $N$ was arbitrary, the normal component of $\Del_{X}Y$, i.e., $II(X, Y)$, depends only on the values $X(p)$ and $Y(p)$.