Locally homogeneous rigid geometric structures on surfaces

Details Locally homogeneous rigid geometric structures on surfaces Locally homogeneous rigid geometric structures on surfaces

2009-07-23

arxiv.org/abs/0907.4072v1

Mathematics

Differential Geometry

21 pages

Scientific paper

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be a unimodular real analytic affine connection on a real analytic compact connected surface $M$. If $\nabla$ is locally homogeneous on a nontrivial open set in $M$, we prove that $\nabla$ is locally homogeneous on all of $M$.

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