Derivative of a $1$-parameter family of Riemannian metrics Let $S^n$ be a closed manifold and let $(M^{n+1},g)$ be a complete Riemannian manifold. Consider $\varphi: S \to M$ a fixed immersion and let $\varphi_t : S \to M$, $t\in(-\varepsilon, \varepsilon)$, be the following $1$-parameter family of immersions (for sufficiently small $\varepsilon$):
$$ \varphi_t(p) = \operatorname{exp}_{\varphi(p)}(t f(p) N(p)), $$
where $f \in C^{\infty}(S)$ and $N$ is a unit normal for $S$ along $\varphi$. How do I compute
$$\left. \frac{\mathrm{d}}{\mathrm{d}t} \right\vert_{t=0} \varphi_t^{\ast}g \quad ?$$
Notice that $\varphi_t^{\ast}g$ is a $1$-parameter family of symmetric $(0,2)$-tensors on $S$. The derivative above resembles the definition of the Lie derivative, but there are no flows of vector fields involved.
 A: In fact, there is a vector field involved, provided we work locally: given $p\in S$, we can choose a neighborhood $U\ni p$ and an $\epsilon>0$ such that the map $\psi:(-\epsilon,\epsilon)\times U\to M$ defined by $\psi(t,x)=\exp(tN(\varphi_0(x))$ is a diffeomorphism onto its image, and thus $\frac{d}{dt}\varphi_t^*g|_{t=0}=\varphi_0^*\mathcal{L}_Vg$, where $V(t,x)=f(x)\partial_t(t,x)$ and $\partial_t$ is the partial derivative w.r.t. the first factor of $(-\epsilon,\epsilon)\times U$ (alternately, the unique local vector field whose restriction to $S$ is $N$ and whose integral curves are geodesics). Both of these vector fields can be locally identified with their pushforwards into $M$ since, $\psi$ is a diffeomorphism.
It turns out this can be related to the extrinsic curvature of $S$. To show this, we can choose two vector fields $X,Y$ on $\varphi((-\epsilon,\epsilon)\times U)$ which are tangent to $S$, and express the Lie derivative in terms of covariant derivatives:
$$\begin{align*}
(\mathcal{L}_{V}g)(X,Y)=&V(g(X,Y))-g([V,X],Y)-g(X,[V,Y]) \\
=&\nabla_V(g(X,Y))-g(\nabla_VX-\nabla_XV,Y)-g(X,\nabla_VY-\nabla_YV) \\
=&g(\nabla_XV,Y)+g(X,\nabla_YV)
\end{align*}$$
Restricting this expression to $U$, we obtain
$$
\left(\frac{\partial}{\partial t}\varphi_t^*g\right)_{t=0}(X,Y)=g(\nabla_X(fN),Y)+g(X,\nabla_Y(fN))=-2fII(X,Y)
$$
where $II$ is the second fundamental form w.r.t. $N$.
