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$\DeclareMathOperator{\vol}{vol}$I've been working through the computation of the first variation of volume presented in Jost's Riemannian Geometry and Geometric Analysis (page 196 in the sixth edition, section titled: Minimal Submanifolds), and I've been getting caught up in all the notation, and I've been having a lot of trouble exactly understanding how to interpret the partial derivatives in this context.

I'll start with the setup: let $M$ be a smooth submanifold of $N$ and let $F:M\times(-\epsilon,\epsilon)\to N$ be a local variation with compact support. For small enough $t$ we have that $\Phi_t(\cdot):=F(\cdot,t)$ is a diffeomorphism from $M\to M_t\subseteq N$. Now let $\{e_1,\dots,e_m\}$ be an orthonormal frame on $M$. Using this diffeomorphism we can write $$\vol(M_t)=\int_{M}\left\langle\Phi_{t*}e_1\wedge\cdots\wedge\Phi_{t*}e_m,\Phi_{t*}e_1\wedge\cdots\wedge\Phi_{t*}e_m\right\rangle^{\frac{1}{2}}\eta_{M},$$ where $\langle{\cdot,\cdot}\rangle$ is the induced inner product on $\bigwedge^m(TN)$, that is $\langle{v_1\wedge\cdots\wedge v_m,w_1\wedge\cdots\wedge w_m}\rangle=\det(\langle{v_i,w_j}\rangle)$, and $\eta_M$ denotes the Riemannian volume form on $M$.

Then we differentiate this with respect to $t$ to find that $$ \left.\frac{d}{dt}\vol(M_t)\right|_{t=0}=\left.\sum_{i=1}^{m}\int_{M}\frac{\left\langle\Phi_{t*}e_1\wedge\frac{\partial}{\partial t}\Phi_{t*}e_i\wedge\cdots\wedge\Phi_{t*}e_m,\Phi_{t*}e_1\wedge\cdots\wedge\Phi_{t*}e_m\right\rangle}{\|\Phi_{t*}e_1\wedge\cdots\wedge\Phi_{t*}e_m\|}\eta_M\right|_{t=0}$$

  • My first question is about the notation $\frac{\partial}{\partial t}\Phi_{t*}e_i$. Should I interpret this as follows: Let $\gamma:(-\epsilon,\epsilon)\to TN$ be given by $\gamma(t)=\Phi_{t*}e_i\in T_{\Phi_{t}(p)}M$. Then does $\frac{\partial}{\partial t}\Phi_{t*}e_i$ simply mean $d\gamma\left(\frac{\partial}{\partial t}\right)$, where we naturally identify $T(T_qN)\cong T_qN$?

He goes on consider the vector field $X:=\left.\frac{\partial}{\partial t}\Phi_{t}\right|_{t=0}$. I'm assuming the interpretation of this vector field is the same as before.

My main confusion is with this next part:

To compute $\frac{\partial}{\partial t}\Phi_{t*}e_i$ at $t=0$ we consider a curve $c_i(s)$ in $M$ with $c_i(0)=p$ and $c_i'(0)=e_i$ and let $c_i(s,t):=\Phi_t(c_i(s))$. Then $$\left.\Phi_{t*}e_i=\frac{\partial}{\partial s}c_i(s,t)\right|_{s=0}.$$

  • How do I justify this? It definitely lives in the right tangent space since $c_i(0,t)=\Phi_t(p)$, but why does this coincide with the pushforward $d\Phi_t(e_i)$. I'm probably missing something pretty fundamental.

Carrying on with the computations we have $$\left.\frac{\partial}{\partial t}\Phi_{t*}e_i\right|_{t=0}=\left.\frac{\partial}{\partial t}\frac{\partial}{\partial s}c_i(s,t)\right|_{s=t=0}=\left.\frac{\partial}{\partial s}\frac{\partial}{\partial t}c_i(s,t)\right|_{s=t=0}=\left.\nabla^N_{\frac{\partial}{\partial s}}X\right|_{s=0} =\nabla^N_{e_i}X, $$ where $\nabla^N$ is the Levi-Civita connection on $N$.

  • My question is why do the partial derivatives commute in this case? Again, how should these mixed partials be understood, and what justifies this computation? Even intuitively, it doesn't make sense to me that they should.
pomegranate
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1 Answers1

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$\def\p{\partial}$For your second point you just need the chain rule: by definition we have $$e_i = c'_i(0) = Dc_i (\frac\p{\p s}\Big|_{s=0}),$$ so since $c_i(\cdot,t) = \Phi_t \circ c_i$ we have $$\frac \p{\p s}\Big|_{s=0}c_i(s,t)=D(\Phi_t \circ c_i)\left(\frac{d}{ds}\right)=D\Phi_t\left(Dc_i(\frac{d}{ds})\right)=D\Phi_t(e_i).$$

For the first and last questions: I think Jost's notation is slightly confusing here, since (as you observed) its most literal interpretation the quantity $$\frac{\p}{\p t}\frac{\p}{\p s}c_i(s,t)$$ should live in the double tangent bundle $T(TM)$, since the base point $\Phi_t (p)$ of the vector $\p c_i / \p s$ varies with $t$. If $\Phi_t (p)$ really is varying in time (i.e. if $X_p \ne 0$) then your $d\gamma(\frac \p {\p t})$ isn't vertical in $T(TM)$, so the identification you propose doesn't make sense - remember that $TM$ has twice the dimension of $M$, so it is only the vertical subspace of $T_v TM$ that we can identify with $T_{\pi(v)} M$.

What is really meant is the covariant derivative along the curve: for example do Carmo would write this as $$\frac{\nabla}{\p t}\frac{\p c_i}{\p s} := \nabla_{\p c_t/\p t} \frac{\p c_i}{\p s}.$$ (do Carmo uses $D$ instead of $\nabla$, which looks less silly.) In fact, from the look of it, Jost switches to a more sensible notation in the next chapter.

The fact that $$\frac{\nabla}{\p t} \frac{\p c_i}{\p s} = \frac{\nabla}{\p s}\frac{\p c_i}{\p t}$$ follows immediately from the symmetry of the Christoffel symbols (i.e. the fact that the connection is torsion-free) when you write it out in coordinates. For example it's Lemma 3.4 in do Carmo's Riemannian Geometry or Prop 44.1 in O'Neill's Semi-Riemannian Geometry (...). If you find yourself not understanding something like this in Jost then I recommend referring to a text more like one of these - Jost gets through a lot of material in that book, so it is quite brief with the basics.

Anthony Carapetis
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