All Questions
Tagged with almost-complex smooth-manifolds
12 questions
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Total space of the frame bundle always an almost complex manifold in $dim >2$?
I seem to remember reading that the total space of the frame bundle of a smooth Riemannian manifold always admits an almost complex structure in dimensions greater than 2.
However now I can't seem to ...
- 352
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Understanding an integrable almost complex structure
My notes only say that
"Definition An almost complex structure is integrable if it is induced by an underlying complex structure."
How do I translate this into a formula? I am not sure why ...
- 3,508
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Existence of almost complex structure on smooth even dimensional manifold
I am trying to prove that if $M$ is an even dimensional manifold, and the bundle of linear frames of $TM$ admits a reduction of its structure to group to $GL_n(\mathbb{C})$, then $M$ admit's an almost ...
- 4,163
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Extension of compatible almost complex structures from a closed set
Suppose that $(M,\omega)$ is a symplectic manifold, and $N \subset M$ is a closed submanifold of $M$. If $J_N: TM|_N \rightarrow TM|_N$ is an $\omega$-compatible almost complex structure, defined on $...
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Almost complex structure on a contractible manifold
Let $M$ be a contractible manifold with an almost complex structure $J:TM\to TM$. Suppose $J':TM\to TM$ is another almost complex structure. Since $M$ is contractible, so is $TM$, hence $J$ and $J'$ ...
- 1,920
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Question about the Newlander-Nirenberg theorem for almost complex manifolds
While studying the definition of almost complex structure, I have thought about almost complex structures which are not complex structures.
To ask a more complete question, I write the following ...
- 915
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The tangent space of an almost complex manifold has a basis of a special form
An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is ...
- 524
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Almost complex structure on real differentiable manifolds
An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is ...
- 524
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Advantage in considering an almost complex structure
Let $M$ be a real Riemannian manifold of even dimension, what are the advantages of considering an almost complex structure on it rather than only real?
user333046
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Complexified tangent bundle
Let $M$ be a smooth manifold and consider the complexified tangent bundle $TM\otimes\mathbb C$. Then there exists a complex-structure $J:TM\otimes\mathbb C\rightarrow TM\otimes\mathbb C$ with $J^2=-I$....
- 4,181
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Almost complex structure on $S^6$
It is known that the sphere $ S^6$ admits an almost complex structure by identifying $S^6 $ with the space of unit purely imaginary Cayley numbers.
I would like to show that this almost complex ...
- 4,181
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smooth vs. analytic in the definition of almost-complex manifolds
Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$,
suppose $\: j : TM\to TM\:$ is a smooth function ...
user57159