# Questions tagged [almost-complex]

For questions about almost complex structures on vector spaces or manifolds.

76 questions
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### Vertical $J$-holomorphic spheres

Let $\pi:M\to N$ be a smooth fiber bundle with fiber $F$ and let $g$ be a Riemannian metric on $M$. Using $g$ we may split the tangent bundle of $M$ as $$TM\cong TF\oplus \pi^{\ast}TN,$$ where $TF$ is ...
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### Equivalence of two definitions of Almost Complex Structure.

Let $M$ be a smooth manifold of even dimension $2n$. I would like to understand the equivalence of the following two standard definitions for an almost complex structure on $M$. Denote by $TM$ the ...
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### $M$ is a complex manifold $\Leftrightarrow \overline{\partial}^2=0$

One way of stating Newlander-Niremberg's theorem is by saying an almost complex manifold $(M,J)$ is complex $\Leftrightarrow \overline{\partial}^2=0$. I'm confused by this, because I can't see why ...
1answer
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### Equivalent definitions of almost complex structures

The definition I have seen for an almost complex structure is the following $$J:TM\to TM$$ which is linear fibre by fibre, such that $J^2 = -\text{Id}$, and such that $\pi(J(X_x)) = x$ where $\pi$ ...
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### Reference for complex structures and Hermitian structures

I am reading Complex Geometry: An Introduction by Daniel Huybrechts. Section on complex structures and Hermitian structures deals with Lefschetz operator denoted by $L$. Hodge * -operator. The ...
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### Curvature identity on Nearly Kähler manifolds

Can somemone help me prove the following identity? $$\| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$ where $J$ is the almost complex structure, and $R$ the curvature ...
1answer
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### Is any 2$m$-dimensional manifold almost complex?

In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write: Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] ...
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### Equivalence of (almost) complex structures

Preamble: An almost complex structure on a manifold $M$ is an endomorphism $J : TM \to TM$ such that $J^2 = -1$. An almost complex structure $J$ is said to be integrable if the Nijenhuis tensor, $N_J$,...
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### Equilateral triangle in complex number [closed]

Let $a$, $b$ and $c$ be the affix of $A$, $B$ and $C$, where $a+b+c=0$ and $a$, $b$ and $c$ are of equal magnitude. Prove that $\triangle ABC$ is an equilateral triangle.
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281 views