Assume that M is a compact 3-dimensional affine manifold which is also a Seifert manifold with vanishing Euler number and whose base orbifold has negative Euler characteristic. How to prove that the locally flat manifold M admits a Hessian metric? Thanks for reponse.
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2$\begingroup$ What is a Hessian metric? Also, does it have to be related to the given (flat?) affine structure? $\endgroup$Moishe Kohan– Moishe Kohan2025-11-01 22:54:38 +00:00Commented Nov 1 at 22:54
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1$\begingroup$ A Riemannian manifold is a Hessian manifold if there exist an atlas of local coordinates where the transition maps between the charts are affine functions and the metric can be written as the Hessian of some convex potential function in each chart. There’s an equivalent definition that such a manifold admits two flat connections $\nabla$ and $\nabla^\ast$ satisfying $g(\nabla_X Y,Z)=g(Y, \nabla^\ast_X Z)$ for any vector fields $X,Y$ and $Z$. $\endgroup$Gabe K– Gabe K2025-11-02 00:20:48 +00:00Commented Nov 2 at 0:20
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3$\begingroup$ Depending on the precise reading of the statement you are asking about (by the way, where did you find it?), the claim is either true or false. (False if the Hessian metric is required to be compatible with the given flat affine structure, true if it is assumed to be compatible with some flat affine structure.) I am voting to close due to lack of clarity. $\endgroup$Moishe Kohan– Moishe Kohan2025-11-03 01:26:42 +00:00Commented Nov 3 at 1:26
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