A Function obstruction to the Existence of Complex Structures

Jun Ling

arXiv:1901.05844·math.GM·March 11, 2019

A Function obstruction to the Existence of Complex Structures

Jun Ling

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TL;DR

This paper introduces a new function that acts as an obstruction to the integrability of almost-complex structures, simplifying the process of determining the existence of complex structures on manifolds.

Contribution

The paper constructs a novel scalar function that serves as an obstruction to complex structure existence, providing an easier alternative to tensor-based methods.

Findings

01

The function is non-zero for non-integrable almost-complex structures.

02

Non-vanishing of the function implies the structure is not integrable.

03

The function simplifies the detection of complex structures.

Abstract

We construct a function for almost-complex Riemannian manifolds. Non-vanishing of the function for the almost-complex structure implies the almost-complex structure is not integrable. Therefore the constructed function is an obstruction for the existence of complex structures from the almost-complex structure. It is a function, not a tensor, so it is easier to work with.

Equations242

N_{J} (X, Y) = [J X, J Y] - J [X, J Y] - J [J X, Y] - [X, Y],

N_{J} (X, Y) = [J X, J Y] - J [X, J Y] - J [J X, Y] - [X, Y],

L_{J} := - \partial_{j} (J_{l}^{i} J_{l}^{k}) \partial_{i} J_{k}^{j} .

L_{J} := - \partial_{j} (J_{l}^{i} J_{l}^{k}) \partial_{i} J_{k}^{j} .

N (X, Z, Y, W)

N (X, Z, Y, W)

:=\frac{1}{4}\Big{\{}\left\langle JN_{J}\Big{(}N_{J}(X,Z),Y\Big{)},W\right\rangle_{g}+\left\langle JN_{J}\Big{(}N_{J}(Y,Z),X\Big{)},W\right\rangle_{g}

:=\frac{1}{4}\Big{\{}\left\langle JN_{J}\Big{(}N_{J}(X,Z),Y\Big{)},W\right\rangle_{g}+\left\langle JN_{J}\Big{(}N_{J}(Y,Z),X\Big{)},W\right\rangle_{g}

+\left\langle JN_{J}\Big{(}N_{J}(X,W),Y\Big{)},Z\right\rangle_{g}+\left\langle JN_{J}\Big{(}N_{J}(Y,W),X\Big{)},Z\right\rangle_{g}\Big{\}}.

+\left\langle JN_{J}\Big{(}N_{J}(X,W),Y\Big{)},Z\right\rangle_{g}+\left\langle JN_{J}\Big{(}N_{J}(Y,W),X\Big{)},Z\right\rangle_{g}\Big{\}}.

g^{ij} g^{k l} N (ε_{i}, ε_{k}, ε_{i}, ε_{k}),

g^{ij} g^{k l} N (ε_{i}, ε_{k}, ε_{i}, ε_{k}),

g^{ij} g^{k l} N (ε_{i}, ε_{k}, ε_{i}, ε_{k}) = L_{J}

g^{ij} g^{k l} N (ε_{i}, ε_{k}, ε_{i}, ε_{k}) = L_{J}

L_{j}:=N(e_{i},e_{k},e_{i},e_{k})=\left\langle JN\Big{(}N(e_{i},e_{k}),e_{i}\Big{)},e_{k}\right\rangle_{g}=N_{ik}^{r}N_{ri}^{s}J_{s}^{k},

L_{j}:=N(e_{i},e_{k},e_{i},e_{k})=\left\langle JN\Big{(}N(e_{i},e_{k}),e_{i}\Big{)},e_{k}\right\rangle_{g}=N_{ik}^{r}N_{ri}^{s}J_{s}^{k},

N_{ij}^{k} = J_{i}^{k} (\partial_{p} J_{j}^{k} - \partial_{j} J_{p}^{k}) - J_{j}^{k} (\partial_{p} J_{i}^{k} - \partial_{i} J_{p}^{k}) .

N_{ij}^{k} = J_{i}^{k} (\partial_{p} J_{j}^{k} - \partial_{j} J_{p}^{k}) - J_{j}^{k} (\partial_{p} J_{i}^{k} - \partial_{i} J_{p}^{k}) .

N_{ik}^{r} N_{r i}^{s} J_{s}^{k}

N_{ik}^{r} N_{r i}^{s} J_{s}^{k}

= {J_{i}^{p} (\partial_{p} J_{k}^{r} - \partial_{k} J_{p}^{r}) - J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})} J_{s}^{k}

= {J_{i}^{p} (\partial_{p} J_{k}^{r} - \partial_{k} J_{p}^{r}) - J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})} J_{s}^{k}

= {J_{s}^{k} J_{i}^{p} (\partial_{p} J_{k}^{r} - \partial_{k} J_{p}^{r}) - J_{s}^{k} J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})}

= {J_{s}^{k} J_{i}^{p} (\partial_{p} J_{k}^{r} - \partial_{k} J_{p}^{r}) - J_{s}^{k} J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})}

= {(J_{s}^{k} J_{i}^{p} \partial_{p} J_{k}^{r} - J_{s}^{k} J_{i}^{p} \partial_{k} J_{p}^{r}) - J_{s}^{k} J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})}

= {(J_{s}^{k} J_{i}^{p} \partial_{p} J_{k}^{r} - J_{s}^{k} J_{i}^{p} \partial_{k} J_{p}^{r}) - J_{s}^{k} J_{k}^{p} (\partial_{p} J_{i}^{r} - \partial_{i} J_{p}^{r})} {J_{r}^{q} (\partial_{q} J_{i}^{s} - \partial_{i} J_{q}^{s}) - J_{i}^{q} (\partial_{q} J_{r}^{s} - \partial_{r} J_{q}^{s})}

= {J_{s}^{k} \cdot J_{i} J_{k}^{r} - J_{i}^{p} \cdot J_{s} J_{p}^{r} + \partial_{s} J_{i}^{r} - \partial_{i} J_{s}^{r}} {J_{r} J_{i}^{s} - J_{r}^{q} \partial_{i} J_{q}^{s} - J_{i} J_{r}^{s} + J_{i}^{q} \partial_{r} J_{q}^{s}}

= {J_{s}^{k} \cdot J_{i} J_{k}^{r} - J_{i}^{p} \cdot J_{s} J_{p}^{r} + \partial_{s} J_{i}^{r} - \partial_{i} J_{s}^{r}} {J_{r} J_{i}^{s} - J_{r}^{q} \partial_{i} J_{q}^{s} - J_{i} J_{r}^{s} + J_{i}^{q} \partial_{r} J_{q}^{s}}

= {J_{s}^{k} \cdot J_{i} J_{k}^{r} - \partial_{i} J_{s}^{r} - J_{i}^{p} \cdot J_{s} J_{p}^{r} + \cdot \partial_{s} J_{i}^{r}} {- J_{i} J_{r}^{s} - J_{r}^{q} \partial_{i} J_{q}^{s} + J_{r} J_{i}^{s} + J_{i}^{q} \partial_{r} J_{q}^{s}}

= {J_{s}^{k} \cdot J_{i} J_{k}^{r} - \partial_{i} J_{s}^{r} - J_{i}^{p} \cdot J_{s} J_{p}^{r} + \cdot \partial_{s} J_{i}^{r}} {- J_{i} J_{r}^{s} - J_{r}^{q} \partial_{i} J_{q}^{s} + J_{r} J_{i}^{s} + J_{i}^{q} \partial_{r} J_{q}^{s}}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} - J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} - J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s}

+ J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} - J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{i} J_{s}^{r}

+ J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} - J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{i} J_{s}^{r}

+ J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s} - J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s}

+ J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s} - J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s}

- J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} - J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r} + J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{s} J_{i}^{r}

- J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} - J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r} + J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{s} J_{i}^{r}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s}

- J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s} - J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} + J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s}

- J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s} - J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} + J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s}

- J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s} - J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r}

- J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s} - J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r}

+ J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} + J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} .

+ J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{i} J_{s}^{r} - J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} + J_{i}^{q} \partial_{r} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} .

- J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} = - \partial_{s} J_{i}^{r} \cdot \partial_{i} J_{q}^{s} \cdot J_{r}^{q} = J_{q}^{s} \cdot \partial_{s} J_{i}^{r} \cdot \partial_{i} J_{r}^{q} = J_{q} J_{i}^{r} \cdot \partial_{i} J_{r}^{q} = J_{j} J_{i}^{k} \cdot \partial_{i} J_{k}^{j} .

- J_{r}^{q} \partial_{i} J_{q}^{s} \cdot \partial_{s} J_{i}^{r} = - \partial_{s} J_{i}^{r} \cdot \partial_{i} J_{q}^{s} \cdot J_{r}^{q} = J_{q}^{s} \cdot \partial_{s} J_{i}^{r} \cdot \partial_{i} J_{r}^{q} = J_{q} J_{i}^{r} \cdot \partial_{i} J_{r}^{q} = J_{j} J_{i}^{k} \cdot \partial_{i} J_{k}^{j} .

- J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} = - J_{j} J_{i}^{k} \cdot \partial_{i} J_{k}^{j}

- J_{r} J_{i}^{s} \cdot \partial_{i} J_{s}^{r} = - J_{j} J_{i}^{k} \cdot \partial_{i} J_{k}^{j}

I I 3 + I V 3 = 0.

I I 3 + I V 3 = 0.

N_{ik}^{r} N_{r i}^{s} J_{s}^{k}

N_{ik}^{r} N_{r i}^{s} J_{s}^{k}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s}

= - J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{i} J_{r}^{s} + J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot J_{r} J_{i}^{s} + J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{i} J_{r}^{s} - J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot J_{r} J_{i}^{s}

- J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s} + I I 3 + J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s}

- J_{r}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{i} J_{q}^{s} + J_{i}^{q} J_{s}^{k} \cdot J_{i} J_{k}^{r} \cdot \partial_{r} J_{q}^{s} + I I 3 + J_{i} J_{r}^{s} \cdot \partial_{i} J_{s}^{r} + J_{r}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{i} J_{q}^{s}

- J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s} - J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r}

- J_{i}^{q} J_{i}^{p} \cdot J_{s} J_{p}^{r} \cdot \partial_{r} J_{q}^{s} - J_{i} J_{r}^{s} \cdot \partial_{s} J_{i}^{r} + J_{r} J_{i}^{s} \cdot \partial_{s} J_{i}^{r}

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TopicsGeometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Geometry and complex manifolds

Full text

An Obstruction Function to the Existence of Complex Structures

Jun LING

Department of Mathematics, Utah Valley University, Orem, Utah 84058

[email protected]

(Date: Dec 14, 2018)

Abstract.

We construct a function for almost-complex manifolds. Non-vanishing of the function for the almost-complex structure implies the almost-complex structure is not integrable. Therefore the constructed function is an obstruction for the existence of complex structures from the almost-complex structure. It is a function, instead of a tensor.

Key words and phrases:

complex structure, obstruction

2000 Mathematics Subject Classification:

53C15, 53A55

1. Introduction

Given $M^{n}$ a smooth manifold, an interesting question is that does there exist any complex manifold structure on $M^{n}$ that makes $M^{n}$ a complex manifold? For a given complex manifold, the underline complex structure gives a canonical almost-complex structure. In the study of existence or nonexistence of complex manifold structure for a manifold, one naturally looks at existence of almost-complex structure first, and should any exists, and check whether or not it can be ”integrated” to a complex structure.

An almost-complex structure $J$ on smooth manifold $M^{n}$ is an endomorphism of the tangent bundle $TM$ with $J^{2}=-1$ . It is known that if $M$ has an almost-complex structure, then $M$ has even dimension $n$ and $M$ is orientable. Nijenhuis tensor $N_{J}$ for the almost complex structure $J$ is given by the following equation

[TABLE]

for all smooth vector fields $X,Y$ . The celebrated Newlander-Nirenberg theorem [1] implies that $N_{J}=0$ if and only if $J$ is a canonical almost-complex structure of a complex manifold. An almost-complex structure $J$ is called integrable if $N_{J}$ vanishes. So in studying existence or nonexistence of complex manifold structure on a manifold, one often studies existence or nonexistence of integrable almost-complex structures, or equivalently studies almost-complex structures and the related Nijenhuis tensor vanishing or non-vanishing property.

On the other hand, though it is great that vanishing or non-vanishing Nijenhuis tensor determines whether the almost-complex structure is integrable or not, it is hard to check the vanishing status for the manifolds other than spheres, due the nature of tensor. It would be nice to give some sufficient condition for existence or to give some obstruction for non-existence, of complex structures. In this paper we give an obstruction that is a function $L_{J}$ , instead of a tensor. Non-vanishing function $L_{J}$ implies non-vanishing Nijenhuis tensor $N_{J}$ . To my knowledge, this is the first of obstruction ever appeared. That opens a door for the research of non-existence of complex structures. We construct the obstruction function in Section 2.

2. An function obstruction for the existence of complex structures

We take the convention that we sum on duplicated index in this paper, unless otherwise stated. Our first result is

Theorem 2.1.

If $J$ is an almost-complex structure on a smooth manifold, and if

[TABLE]

is not zero at some point of $M$ , Then $J$ is not integrable.

Proof.

First take a metric $g$ on the smooth manifold $M$ that always exists. ( $L_{J}$ is independent of $g$ and is in terms of almost-complex structure itself in simple form). We define a $(4,0)$ tensor $N(X,Z,Y,W)$ by

[TABLE]

where $N_{J}$ is the one in (1.1) and $\langle\cdot,\cdot\cdot\rangle_{g}=g(\cdot,\cdot\cdot)$ .

Take the trace of the first argument and the third argument and then take the trace of the second and the fourth argument at a point to get a number for that point

[TABLE]

where $\{\varepsilon_{i}=\frac{\partial}{\partial x^{i}}\}_{i=1}^{n}$ is a local frame of local coordinate system $\{x^{i}{i}\}_{i=1}^{n}$ , $g_{ij}=g(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial x^{j}})$ , matrix $(g^{ij})=(g_{ij})^{-1}$ . We write $\partial_{i}=\frac{\partial}{\partial x^{i}}$ , $\nabla_{i}=\nabla_{\frac{\partial}{\partial x^{i}}}$ $\langle\cdot,\cdot\cdot\rangle_{g}=g(\cdot,\cdot\cdot)$ for convenience.

If we can show

[TABLE]

then $L_{J}$ is constructed from Nijenhuis tensor $N_{J}$ by contractions. Non-vanishing function $L_{J}$ implies non-vanishing tensor $N_{J}$ since $N_{J}=0$ implies $L_{J}=0$ . Therefore the theorem follows.

We now prove (2.2).

We calculate at a point with normal coordinates. So at the point $g_{ij}=\delta_{ij}$ , $\delta_{ii}=1$ , $\delta_{ij}=0$ if $i\not=j$ .

Note that $N(X,Z,X,Z)=\left\langle JN_{J}\Big{(}N_{J}(X,Z),X\Big{)},Z\right\rangle_{g}$ . therefore

[TABLE]

where $N_{ij}^{k}$ are given by $N_{J}(\partial_{i},\partial_{j})=N_{ij}^{k}\partial_{k}$ . It is easy to see

[TABLE]

We write $J_{i}:=J_{i}^{j}\partial_{j}$ for convenience.

[TABLE]

IV3, meaning the third term of fourth line in the above four lines after $=$

[TABLE]

II3, meaning the third term of fourth line in the four lines after $=$ . Others are similar.

[TABLE]

IV2

[TABLE]

II5

[TABLE]

II2

[TABLE]

III2

[TABLE]

III1

[TABLE]

II1

[TABLE]

II4

[TABLE]

IV4

[TABLE]

III3

[TABLE]

I3

[TABLE]

I4

[TABLE]

The first term above is zero. In fact,

[TABLE]

Therefore

[TABLE]

Now

[TABLE]

The sum of the last two terms is zero, for

[TABLE]

Therefore

[TABLE]

∎

Bibliography1

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds , Ann. of Math., 65 (1957), 391–404.

TL;DR

Contribution

Findings

Abstract

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An Obstruction Function to the Existence of Complex Structures

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

2. An function obstruction for the existence of complex structures

Theorem 2.1**.**

Proof.

Theorem 2.1.