material derivative and relation to Riemannian metric

Question

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.

Answer 1

Either you were told wrong or your memory is incomplete

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

2. 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 $\partial_t (\phi^* u) = D_t u = \partial_t u + v\cdot \nabla u$.

3. 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.

4. 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".