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C.F.G
C.F.G
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Almost Complex manifolds of constant curvature

Edited (after R. Bryant comment)

Let $(M,\cal J,g)$ be a almost Hermitian manifold (not necessary integrable). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be aany local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$ where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Complex manifolds of constant curvature

Let $(M,\cal J,g)$ be a Hermitian manifold. i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be a local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$ where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Almost Complex manifolds of constant curvature

Edited (after R. Bryant comment)

Let $(M,\cal J,g)$ be a almost Hermitian manifold (not necessary integrable). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be any local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$ where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

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C.F.G
C.F.G
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Let $(M,\cal J,g)$ be a Hermitian manifold. i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be a local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$

Then where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Let $(M,\cal J,g)$ be a Hermitian manifold. i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be a local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$

Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Let $(M,\cal J,g)$ be a Hermitian manifold. i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be a local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$ where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Source Link
C.F.G
C.F.G
  • 4.2k
  • 6
  • 31
  • 65

Complex manifolds of constant curvature

Let $(M,\cal J,g)$ be a Hermitian manifold. i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be a local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$

Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.