All Questions
7 questions
2
votes
0
answers
81
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Restriction of an almost-complex structure to a complex structure on a sub-manifold?
I have been thinking about this recent question of mine a bit more and came to the following question: Consider a manifold $M$ endowed with a non-integrable almost complex structure $J$. Can it happen ...
1
vote
0
answers
112
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Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
- 103
0
votes
1
answer
255
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Torsion free Chern connections and Kähler manifolds
Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
- 415
2
votes
1
answer
483
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Curvature forms of holomorphic line bundles
Let $M$ be a compact complex manifold, $L$ a holomorphic line bundle over $M$, and $\nabla$ a connection extending the holomorphic structure map $\overline{\partial}$ of $L$. In general can it happen ...
- 969
1
vote
0
answers
162
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Warped product manifold with real and complex parts
Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
- 272
4
votes
1
answer
1k
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Confusion about complex differential forms
I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...
- 143
6
votes
1
answer
518
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Where do the (Akizuki)-Nakano Identities First Appear
The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...
