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I am currently reading a paper about elastic energy of non euclidean thin bodies, about which I might want to write my thesis for my bachelors degree.
You can find it here: https://arxiv.org/abs/1801.02207
While I was reading, I stumbled upon a distance function in the following context, which I do not fully understand:
Given an $n$-dimensional compact, oriented riemannian manifold $(M,g)$, they define the "Hookean energy" as $$E_M:W^{1,2}(M;\mathbb R^n)\rightarrow\mathbb R$$ $$E_M[u]=\frac{1}{\operatorname{Vol}(M)}\int_M\operatorname{dist}^2(du,\operatorname{SO}(g,e))d\operatorname{Vol}_g$$ where $e$ is the euclidian metric and $\operatorname{SO}(g,e)$ is the set of orientation preserving isometries $T_p M\rightarrow\mathbb R^n$.
Then the distance is measured w.r.t. the inner product norm on $T_p^* M\otimes\mathbb R^n$.
They then say that in an orthonormal basis at $T_p M$, $\operatorname{SO}(g,e)_p$ and dist reduce to $\operatorname{SO}(n)$ and the Frobenius distance.

Can someone define this distance function explicitly for me please?
And just to clarify, $du:TM\rightarrow\mathbb R^n$ is just the tangent map, right? Because it was confusing to me, as $d$ is also used for the De Rham differential.

PS: Can anyone recommend a book on this topic?

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$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\SO}{\operatorname{SO}(V,W)}$

I am one of the authors of the paper that you mention:)

The first point is that this is actually a pointwise issue: For $p \in M$, $df_p:(T_pM,g_p ) \to (\mathbb{R}^n,e)$ is a linear map between two inner product spaces.

So, this is a particular instance of the following situation: We have two $n$-dimensional oriented inner product spaces $V,W$ and a linear map $T \in \Hom(V,W)$. We want to define $ \operatorname{dist}(T,\operatorname{SO}(V,W))$. The inner products on $V,W$ induce an inner product on $\Hom(V,W)$, which makes $\Hom(V,W)$ into a metric space. Then $$ \operatorname{dist}(T,\operatorname{SO}(V,W)):=\inf_{Q \in \SO}d(T,Q),$$

i.e. this is just the usual distance of an element in a metric space from a subset.


Now, there are various equivalent ways to see how the inner products on $V,W$ induce an inner product on $\Hom(V,W)$:

  1. The most concrete approach is to choose (positively-oriented) orthonormal bases for $V,W$. Given such bases, the inner product on $\Hom(V,W)$ is defined to be $$ \langle T,S \rangle:=\langle [T],[S] \rangle,$$ where $ [T],[S]$ are the representing matrices of $T,S$ w.r.t the chosen bases, and $\langle [T],[S] \rangle$ is merely the standard Frobenius inner product between matrices. (Of course, one needs to check that this definition does not depend on the bases chosen, as long as they are orthonormal). The map $T \to [T]$ maps $\SO$ to $\operatorname{SO}(n)$, and we have $\operatorname{dist}(T,\operatorname{SO}(V,W))=\operatorname{dist}([T],\operatorname{SO}(n))$. (This should explain the "reduction" comment in the paper).

  2. Another approach is to define $ \langle T,S \rangle=\text{tr}(S^T \circ T)$. Note that $S^T \circ T \in \Hom(V,V)$ so its trace is well-defined.

  3. One can identify $\Hom(V,W) \cong V^* \otimes W$. The product on $V$ induces a product on $V^*$, and now the products on $V^*,W$ induce a product on the tensor product $V^* \otimes W$. (see here for instance).

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