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For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group.

Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = \{~\phi \in \mathrm{Diff}(M) : \phi^*(\mathcal{K}) = \mathcal{K}~\}.$$

Some special cases are:

  • If $\mathcal{K}$ is a metric, then $\mathrm{Diff}_{~\mathcal{K}}(M)$ is just the isometry group of $\mathcal{K}$. It is well-known that this is a finite-dimensional Lie group. In some cases, this group might even be trivial, consisting of the identity map alone.
  • On the other hand, if $\mathcal{K} = 0$, then $\mathrm{Diff}_{~\mathcal{K}}(M) = \mathrm{Diff}(M)$, which is not a (finite-dimensional) Lie group.
  • The previous example might seem too extravagant, but even if $\mathcal{K}$ is a non-degenerate $2$-form, it might happen that $\mathrm{Diff}_{~\mathcal{K}}(M)$ is still too big to be a (again, finite-dimensional) Lie group. This is what happens with symplectic forms, for instance.

These examples show that this group depends quite a bit on conditions we impose on $\mathcal{K}$, and it's not clear to me exactly how this happens, or why. So, my questions are:

1) Is there a general theory that says under which conditions (on $\mathcal{K}$) can we expect this group to be finite-dimensional?

2) Is there any 'big picture' explanation for this phenomenon?


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You should modify your title because a metric is not 2-form! – Mercy King Jun 30 '12 at 1:56
@Mercy I guess you're right.. I meant 2-tensor. Thanks. – student Jun 30 '12 at 2:09
Is there any particular reason to restrict this question to $(0,2)$-tensors? – Neal Jun 30 '12 at 2:21
@Neal It seems to me that $2$-tensors deserve a special place among the others. But of course, the same question goes for any tensor. – student Jul 1 '12 at 17:20

S.Kobayashi, "Transformation groups in differential geometry",

discusses G-structures (2-tensors are special cases) and when their automorphism groups are finite dimensional Lie groups. The key is the "elliptic" condition, Theorem 4.1 (Chapter I). This theorem deals with the case of compact manifolds, but, I think, this condition is irrelevant in the case of G-structures given by 2-tensors. One special case is that of pseudo-Riemannian manifolds: The automorphism group is a finite dimensional Lie group.

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