# Affinely flat structures. How many different ones on the same manifold?

Let $M$ be a smooth manifold endowed with $\nabla$ a flat torsion-free connection. Let $\operatorname{Diff}(M)$ be the group of smooth diffeomorphisms of $M.$ Obviously if $\phi \in \operatorname{Diff}(M)$, then $\phi^* \nabla$ is torsion-free and flat, but equivalent (through $\phi$) to $\nabla.$

Is it known whether the set of nonequivalent affinely flat connections is infinite (if it is not empty)?. In the case of the $2$-dimensional torus it is known that the set of flat, torsion free connections quotient out by the group of diffeomorphisms is a $4$-dimensional manifold.

• Ali, "torsion free" is also a typo and should be "torsion-free" (and also, "$k$-dimensional" with a dash), and "quotiented out" – YCor Mar 1 '16 at 11:27

The set of affinely flat structures has been studied by many authors. Among them, we can quote William Goldman who has studied the deformations of $(X,G)$-structures. Let $X$ be a manifold, $G$ a Lie group which acts on $X$, we suppose that if two elements of $G$ coincide on an open subset of $X$, they coincide on $X$. A manifold $M$ is an $(X,G)$-manifold if it is endowed with an atlas $(U_i,\phi_i)$, where $\phi_i\colon U_i\subset M\rightarrow X$ is a diffeomorphism between $U_i$ and its image. We suppose that $\phi\circ {\phi_j}^{-1}$ is the restriction of an element of $G$. A flat torsionless connection defined on the $n$-dimensional manifold $M$ defines on it a $(\mathbb{R}^n,\operatorname{Aff}(\mathbb{R}^n))$-structure.

The $(X,G)$-structure of $M$ can be lifted to its universal cover $\tilde M$, and there exists a developing map: $D\colon\tilde M\rightarrow X$ such that for every element $\gamma\in\pi_1(M), D\circ \gamma =h(\gamma)\circ D$. We thus have a representation $h\colon\pi_1(M)\rightarrow G$ called the holonomy representation of the $(X,G)$-structure on $M$. It is a result of Thurston that if $M$ is closed the set of holonomy of $(X,G)$-structures defined on $M$ is an open subset of $\operatorname{Rep}(\pi_1(M),G)$, the set of representations $h:\pi_1(M)\rightarrow G$.

Goldman, William M. "Varieties of representations." Geometry of Group Representations: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held July 5-11, 1987 with Support from the National Science Foundation. Vol. 74. American Mathematical Soc., 1988.

Available here:

http://www-users.math.umd.edu/~wmg/geost.pdf

For the dimension 2, a theorem of Benzecri shows that an oriented surface is endowed with an affine structure if and only if its genus is one.

• It could also be helpful to note that the map assigning to an $(X,G)$-structure its holonomy is a local homeomorphism on moduli, so the $(X,G)$-structure moduli space is locally homeomorphic to an open subset of the representation variety of the fundamental group. – Ben McKay Mar 1 '16 at 10:33