Understanding the space of structures

Question

Apologies if this question is a little vague.

I have seen written in a few places that the space of projective structures on a Riemann surface is an affine space modelled on the space of holomorphic quadratic differentials. Firstly, is this true for real curves and higher dimensional projective structures? Secondly, is there an analogous result for other Cartan-geometric structures (or G-structures) that the space of such structures is an affine space modelled on some appropriate vector bundle?

Any references or pointers in the right direction greatly appreciated.

Answer 1

The space of affine connections is an affine space, since any two affine connections, say with Christoffel symbols $\Gamma^i_{jk}$ and $\bar\Gamma^i_{jk}$ differ by the difference of their Christoffel symbols $\Gamma^i_{jk}-\bar\Gamma^i_{jk}=a^i_{jk}$, so the difference determines a tensor $a^i_{jk}dx^j dx^k\partial_i$. If they are torsion free this tensor will be symmetric. So both the moduli space of affine connections and of affine torsion free connections are affine spaces, modelled on $T^* \otimes T^* \otimes T$ and $S^2T^* \otimes T$ respectively.

Every affine connection has the same unparameterized geodesics as its associated torsion free connection with symmetrized Christoffel symbols $\bar\Gamma^i_{jk}=(1/2)(\Gamma^i_{jk}+\Gamma^i_{kj})$ (proof: write out the geodesic equations). Every torsion free connection with Christoffel symbols $\Gamma^i_{jk}$ has the same unparameterized geodesics as another torsion free connection just when the second has Christoffel symbols $\Gamma^i_{jk}+a_j\delta^i_k+a_k\delta^i_j$ for any $1$-form $a_idx^i$, in any local coordinates. So the space of normal projective connections is the quotient space of the space of affine connections, which we can assume to be torsion free, by the space of $1$-forms. If they are torsion free with the same geodesics, it will be of this form $a^i_{jk}=a_j\delta^i_k+a_k\delta^i_j$. So the space of normal projective connections is therefore an affine space modelled on the quotient space, i.e. on the affine space of traceless symmetric tensors $a^i_{jk}dx^j dx^k\partial_i$, $a^i_{jk}=a^i_{kj}$, $a^i_{ik}=0$, which we can denote $T \otimes_0 S^2 T^*$, or something like that. This discussion works the same for the real normal projective connections and for the holomorphic ones, so we see that the moduli space of holomorphic normal projective connections on any compact complex manifold is finite dimensional, modelled on the vector space $H^0(M,T \otimes_0 S^2 T^*)$.

On a complex manifold, a holomorphic affine connection exists just when the Atiyah class of the tangent bundle vanishes. Less obviously: a holomorphic projective connection exists just when a normal holomorphic projective connection exists, both just when the tracefree part of the Atiyah class of the tangent bundle vanishes.

Answer 2

This result is true for general $\mathbb{R}P^n$ manifolds. A $\mathbb{R}P^n$-manifold, is a manifold endowed with an atlas $(U_i,f_i)$ such that $f_i:U_i\rightarrow \mathbb{R}P^n$ is a diffeomorphism onto its image, and $f_i\circ f_j^{-1}$ is the restriction of a projective map.

A $\mathbb{R}P^n$-manifold is also defined by a projectively flat connection. W. Goldman has shown that the space of projective structure defined on $M$ is affine space whose underlying vector subspace is $\Omega^1(M)$.

Goldman, W. The symplectic geometry of affine connections on surfaces. p.135