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Mike Cocos
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Are compact,complex and affinnely flat manifolds geodesically complete

Let M be a complex manifold with real tangent bundle TM. Let $J$ be its associated almost complex structure($JoJ=-id$) and $\nabla$ a torsion free, flat connection in $TM$ compatible with $J$, that is $$\nabla J=0.$$ Is it true that in this case the manifold is geodesically complete?

If one replaces the complex structure by a simplectic structure the question becomes a simpler version of Markus conjecture.

Mike Cocos
  • 463
  • 3
  • 10