Lower order tensors in non-Kähler geometry and non-Kähler geometric flow

Abstract

In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct the second-order curvature flows in a uniform way, under some natural assumptions which hold in Streets and Tian’s works. As a result, by classifying the lower order tensors, we classify the second-order curvature flows in almost Hermitian, almost Kähler, and Hermitian geometries in certain sense. In particular, the Symplectic Curvature Flow is the unique way to generalize Ricci Flow on almost Kählermanifolds.

Appendix: Hermitian connection

In this appendix, we review some basic results about Hermitian connection. For further study, one may refer [9].

Let $(g,J,\omega )$ be an almost Hermitian structure. Let D be Levi-Civita connection and $\bigtriangledown$ be a linear connection. Let $\bigtriangledown =D+A$, i.e., $g(\bigtriangledown _{X}Y,Z)=g(D_{X}Y,Z)+A(X,Y,Z)$, where A is a 3-tensor.

Lemma 6.1

$$\begin{aligned} \bigtriangledown g=0\Leftrightarrow & {} A(X,Y,Z)+A(X,Z,Y)=0.\\ \bigtriangledown J=0\Leftrightarrow & {} A(X,JY,Z)+A(X,Y,JZ)+DJ(X,Y,Z)=0. \end{aligned}$$

Proof

$$\begin{aligned} \nabla g(X,Y,Z)= & {} Xg(Y,Z)-g(\nabla _{X}Y,Z)-g(Y,\nabla _{X}Z)\\= & {} Xg(Y,Z)-g(D_{X}Y,Z)-A(X,Y,Z)-g(Y,D_{X}Z)-A(X,Z,Y)\\= & {} -\,A(X,Y,Z)-A(X,Z,Y).\\ g(\nabla J(X,Y),Z)= & {} g(\nabla _{X}(JY)-J\nabla _{X}Y,Z)\\= & {} g(D_{X}(JY),Z)+A(X,JY,Z)-g(JD_{X}Y,Z)+A(X,Y,JZ)\\= & {} DJ(X,Y,Z)+A(X,JY,Z)+A(X,Y,JZ). \end{aligned}$$

$\square$

If $\bigtriangledown g=\bigtriangledown J=0$, then, we say that $\bigtriangledown$ is an Hermitian connection. In general, Hermitian connection is not unique. Naturally, we assume A is defined from $(g,J,\omega )$. Since connection is of first order, we require A is of first order. In addition, since D is of even type, we require A is of even type. In a word, we assume $A=J*DJ$.

Lemma 6.2

Let $\bigtriangledown$ be an Hermitian connection.

In almost Hermitian setting,

$$\begin{aligned} A=\frac{1}{2}DJ(X,JY,Z)+\frac{t}{4}(DJ(JY,Z,X)+DJ(JZ,X,Y)-DJ(Y,Z,JX)-DJ(Z,X,JY)). \end{aligned}$$

In Hermitian setting,

$$\begin{aligned} A=\frac{1}{2}DJ(X,JY,Z)-\frac{t}{2}(DJ(Y,Z,JX)+DJ(Z,X,JY)). \end{aligned}$$

In almost Kählersetting,

$$\begin{aligned} A=\frac{1}{2}DJ(X,JY,Z). \end{aligned}$$

Proof

Suppose $(g,J,\omega )$ is an almost Hermitian structure. From our assumption and Lemma 2.5, we have

$$\begin{aligned} A(X,Y,Z)= & {} a_{1}DJ(X,Y,JZ)+a_{2}DJ(Y,Z,JX)+a_{3}DJ(Z,X,JY)\\&+a_{4}DJ(JX,Y,Z)+a_{5}DJ(JY,Z,X)+a_{6}DJ(JZ,X,Y). \end{aligned}$$

From Lemma 6.1, $\nabla$ is Hermitian if and only if

$$\begin{aligned} 0= & {} a_{1}DJ(X,Y,JZ)+a_{2}DJ(Y,Z,JX)+a_{3}DJ(Z,X,JY)\\&+\,a_{4}DJ(JX,Y,Z)+a_{5}DJ(JY,Z,X)+a_{6}DJ(JZ,X,Y)\\&+\,a_{1}DJ(X,Z,JY)+a_{2}DJ(Z,Y,JX)+a_{3}DJ(Y,X,JZ)\\&+\,a_{4}DJ(JX,Z,Y)+a_{5}DJ(JZ,Y,X)+a_{6}DJ(JY,X,Z).\\ -DJ(X,Y,Z)= & {} a_{1}DJ(X,JY,JZ)+a_{2}DJ(JY,Z,JX)-a_{3}DJ(Z,X,Y)\\&+\,a_{4}DJ(JX,JY,Z)-a_{5}DJ(Y,Z,X)+a_{6}DJ(JZ,X,JY)\\&-\,a_{1}DJ(X,Y,Z)+a_{2}DJ(Y,JZ,JX)+a_{3}DJ(JZ,X,JY)\\&+\,a_{4}DJ(JX,Y,JZ)+a_{5}DJ(JY,JZ,X)-a_{6}DJ(Z,X,Y). \end{aligned}$$

To simplify the above equations, we have

$$\begin{aligned} 0= & {} (a_{2}-a_{3})(DJ(Y,Z,JX)-DJ(Z,X,JY))+(a_{5}-a_{6})(DJ(JY,Z,X)-DJ(JZ,X,Y)).\\ 0= & {} (1-2a_{1})DJ(X,Y,Z)+2a_{4}DJ(JX,JY,Z)\\&+(a_{2}+a_{5})(DJ(JY,Z,JX)-DJ(Y,Z,X))+(a_{3}+a_{6})(DJ(JZ,X,JY)-DJ(Z,X,Y)). \end{aligned}$$

Therefore, in almost Hermitiansetting, $a_{1}=\frac{1}{2}$, $a_{4}=0$, $a_{2}=a_{3}=-a_{5}=-a_{6}=-\frac{t}{4}$.

In Hermitian setting, from Lemma 2.6,

$$\begin{aligned} A(X,Y,Z)=(a_{1}-a_{4})DJ(X,Y,JZ)+(a_{2}-a_{5})DJ(Y,Z,JX)+(a_{3}-a_{6})DJ(Z,X,JY), \end{aligned}$$

and the equations are

$$\begin{aligned} 0= & {} (a_{2}-a_{5}-a_{3}+a_{6})(DJ(Y,Z,JX)-DJ(Z,X,JY))\\ 0= & {} (1-2a_{1}+2a_{4})DJ(X,Y,Z). \end{aligned}$$

Therefore, $a_{1}-a_{4}=\frac{1}{2}$, $a_{2}-a_{5}=a_{3}-a_{6}=-\frac{t}{2}$.

In almost Kählersetting, from Lemma 2.7,

$$\begin{aligned} A(X,Y,Z)=(a_{1}+a_{4})DJ(X,Y,JZ)+(a_{2}+a_{5})DJ(Y,Z,JX)+(a_{3}+a_{6})DJ(Z,X,JY), \end{aligned}$$

and the equations are

$$\begin{aligned} 0= & {} (a_{2}+a_{5}-a_{3}-a_{6})(DJ(Y,Z,JX)-DJ(Z,X,JY))\\ 0= & {} (1-2a_{1}-2a_{4})DJ(X,Y,Z)-(2a_{2}+2a_{5})DJ(Y,Z,X)-(2a_{3}+2a_{6})DJ(Z,X,Y). \end{aligned}$$

Therefore, $a_{2}+a_{5}=a_{3}+a_{6}$, $a_{1}+a_{4}=a_{2}+a_{5}+\frac{1}{2}$. Then

$$\begin{aligned} A(X,Y,Z)=\frac{1}{2}DJ(X,Y,JZ). \end{aligned}$$

$\square$