material derivative and relation to Riemannian metric For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$ have the material derivative
$$Du = \tilde u_t + \nabla \tilde u \cdot V$$
where $V$ is the velocity of the hypersurface. Here $\tilde u$ is some extension of $u$ onto an open set around the hypersurface, and $\tilde u_t$ is the ordinary time derivative.
I was recently told that this material derivative is related to differentating the metric in a Riemannian manifold. Could someone give me some detail on this or refer to me a text that discusses this notion? Thanks.
 A: Either you were told wrong or your memory is incomplete 


*

*All geometric notions of derivatives coincide on smooth functions. The metric doesn't enter into it. 

*Your summary of what a material derivative is is not quite correct/complete. What you should have is some version of the following:
Let $M$ denote the material manifold, which is just some $m$ dimensional smooth manifold. Let $N$ denote the spatial manifold, which is some $n$ dimensional smooth manifold. The material manifold is traced out in $\mathbb{R}\times N$ by a (smooth) family of embeddings $\phi(t,\cdot): M \to \{t\} \times N$. 
Let $u:\mathbb{R}\times N\to \mathbb{C}$ be a smooth function. We can compute $\partial_t u$ using the product structure of $\mathbb{R}\times N$ as the derivative along the "$\mathbb{R}$" direction. This is the time derivative of $u$ in the "Eulerian" picture. 
The material derivative, or the "time derivative in the Lagrangian picture", can be expressed as $\partial_t (\phi^* u)$, where $\phi^* u$ is the pullback of $u$ or more precisely the function $\phi^* u: \mathbb{R}\times M \to \mathbb{C}$ is given by $\phi^*(u)(t,x) = u(\phi(t,x))$. Performing the change of variables we see that, decomposing vectors again using the product structure of $\mathbb{R}\times N$, we have that $\partial_t \phi = (1,v)$ where $v$ is some vector tangent to $N$. And so we can write $\partial_t (\phi^* u) = D_t u = \partial_t u + v\cdot \nabla u$.  

*Now instead of $u$ being a smooth function, you can let $u$ be any smooth section of some vector bundle $F$ over $\mathbb{R}\times N$. Then given a connection $\nabla$ of $F$, we can define also the Eulerian time derivative $\nabla_{\partial_t} u$. And similarly we can define the material time derivative as $D_t u = \nabla_{\partial_t \phi} u$. The connection and geometry enters because you are dealing with vector bundles, and you still don't need a metric. 

*Perhaps the intended interpretation of what is described is simply the following: 


*

*We understand the phrase "Riemannian manifold" to be "some submanifold of Euclidean space $\mathbb{R}^n$. 

*Then the emphasis is on the following fact:
Let $M$ be a submanifold of $\mathbb{R}^n$. Let $(y_1, \ldots, y_m)$ be a local coordinate chart of $M$. Suppose $y_1 = x_1$, where $(x_1, \ldots, x_n)$ is the standard coordinates of $\mathbb{R}^n$. Let $u:\mathbb{R}^n\to\mathbb{R}$ be a function. Then it is not true that 
$$ \partial_{x_1} u = \partial_{y_1} u|_M \tag{B}$$
where on the left it is interpreted as the partial derivative of $u$ relative to the coordinate $x_1$ in the standard coordinate system of $\mathbb{R}^n$, and on the right it is interpreted as the partial derivative of $u|_M$ relative to the coordinate $y_1$ in the coordinate system $(y_1, \ldots, y_m)$. 
Performing the correct change of variables you will see that the correct relation is actually
$$ \partial_{y_1} u|_M = \partial_{x_1} u + \underbrace{\sum_{j = 2}^n \frac{\partial x_j}{\partial y_1} \partial_{x_j} u}_{= v \cdot \nabla u} $$
The frequency at which students mistakenly believe that (B) holds has led some people to jokingly refer to that as the "first fundamental mistake of multivariable calculus". 
