# When is the Lie algebra of automorphisms of a geometrical structure finite-dimensional?

Let $$M$$ be an $$n$$-dimensional smooth manifold and $$\Theta$$ some tensor field on $$M$$, so a smooth section of $$TM^{\otimes r} \otimes T^*M^{\otimes s}$$ for some $$(r,s)$$. Let $$\mathfrak{g}_\Theta$$ denote the Lie subalgebra of vector fields which leave $$\Theta$$ invariant: $$\mathfrak{g}_\Theta = \{ X \in \mathcal{X}(M) \mid \mathcal{L}_X \Theta = 0 \}$$

If $$\Theta = g$$, a pseudo-riemannian metric, then it is well-known that $$\mathfrak{g}_\Theta$$ is finite-dimensional, with dimension bounded above by $$n(n+1)/2$$. On the other hand if $$\Theta = \omega$$, a symplectic structure, then it is again well known that $$\mathfrak{g}_\Theta$$ is infinite-dimensional, since it contains the hamiltonian vector fields.

I expect it is not easy, given $$\Theta$$, to determine whether $$\mathfrak{g}_\Theta$$ is finite- or infinite-dimensional, but I thought I'd ask here.

I'm actually interested in knowing what the generic case is.

I know from examples that if we allow a metric to become degenerate, then the "isometry" Lie algebra becomes infinite-dimensional. But on the other hand, a random metric need not have any isometries at all. Hence I am not sure which way to bet.

The general procedure for deciding the 'local' version of this question, at least in the real-analytic connected case, was certainly known to Élie Cartan and, probably known to Lie in some form. The basic idea is this: The condition that a vector field $$X$$ preserve $$\Theta$$ is a system $$\Sigma_\Theta$$ of linear partial differential equations on $$X$$. $$\Sigma_\Theta$$ may not be involutive, in which case, one prolongs the system to a system $$\Sigma'_\Theta$$ that is involutive and then one computes the Cartan characters. If the last nonzero character is $$s_p$$ where $$p>0$$, then the sheaf of local vector fields that satisfy $$\Sigma_\Theta$$ has infinite dimensional stalks and hence the local infinitesimal symmetries of $$\Theta$$ are an infinite dimensional Lie algebra. If the last nonzero character is $$s_0$$, then the sheaf of local vector fields that satisfy $$\Sigma_\Theta$$ is a finite dimensional Lie algebra of dimension $$s_0$$.

The assumption of real-analyticity is not always needed. The global question (i.e., how many global solutions $$X$$ there are) is more subtle, and the answer often depends on details in the particular case, as usual.

There is often a way to show that the end result is going to be finite dimensional before you actually have to compute the involutive prolongation $$\Sigma'_\Theta$$, and such 'practical' criteria are more often used than the full result.

If an analytic geometric structure is rigid in the sense of Gromov (see Gromov and d'Ambra), its symmetry vector fields form a finite dimensional Lie algebra. This is not very helpful, since we don't know too many different ways to prove rigidity, but there are lots of examples in:

D'Ambra, G.(F-IHES); Gromov, M.(I-CAGL) Lectures on transformation groups: geometry and dynamics. Surveys in differential geometry (Cambridge, MA, 1990), 19–111, Lehigh Univ., Bethlehem, PA, 1991.

For example, if a structure (for example, a pseudo-Riemannian metric) induces an affine connection (for example, the Levi--Civita connection), then it is rigid, and its Lie algebra of symmetry vector fields is finite dimensional. Similarly for a projective connection, or (in dimension 3 or more) for a conformal connection.

You could also look at:

M. Gromov, Rigid transformations groups, in Géométrie Différentielle (Paris, 1986), Hermann, 1988.

A. M. Amores, Vector fields of a finite type G-structure, J. Diff. Geom., 1979.

R. Quiroga-Barranco; A. Candel, Rigid and finite type geometric structures. Geom. Dedicata 106 (2004), 123–143.

R. Zimmer, Ergodic theory and the automorphism group of a $$G$$-structure, in Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984), 1987.

The paper of Quiroga-Barranco and Candel explains how to prove rigidity of many types of geometric structures, including Cartan geometries modelled on effective homogeneous spaces, so including pseudo-Riemannian geometries, affine connections, conformal connections in dimension 3 or more, and projective connections in dimensions 2 or more.

• Could you maybe provide me with a more precise pointer to where your last paragraph is in the reference you mention? – ungerade Nov 5 at 0:08
• p. 69 5.11.B (ii): affine structures, (iii): conformal structures. I don't think they cover projective connections, but the rigidity in Gromov's sense of Cartan connections modelled on effective homogeneous spaces is well known, and rigidity for projective connections is proven in Amores, Vector fields ..., JDG, 1979, but was certainly known to Cartan. – Ben McKay Nov 5 at 8:52

This article gives many examples of manifolds endowed with a structure whose group of automorphisms is finite dimensional.

https://www.ams.org/journals/tran/1964-113-01/S0002-9947-1964-0164299-4/S0002-9947-1964-0164299-4.pdf

In this article, it is shown that the group of automorphisms of an elliptic $$G$$-structure defined on a compact manifold is finite dimensional.