Vanishing of the Nijenhuis tensor












2














The Nijenhuis tensor is defined to be:



$$(1):quad
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],
$$



for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:



$$(2):quad
J:TMrightarrow TMquad|quad J^2=-I_{TM}.
$$



The requirement is to show that given $J$ is integrable, $N_J$ is vanishing. I am having an issue with the following computation which gives a vanishing $N_J$, regardless of whether $J$ is integrable or not:



$$
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]
$$

$$
=[X,Y]+J^2XY-JYJX+JXJY-J^2YX-[JX,JY]
$$

$$
=[X,Y]+J^2[X,Y]
$$

$$
=[X,Y]-[X,Y]=0quadtext{(by (2) alone)}.
$$

I find this very strange because this makes absolutely no reference at all to $J$ being integrable or not, rather this follows solely from the definition of $J$ in $(2)$. Suggesting that if $J$ is an almost complex structure, then $N_J$ is always vanishing. Is it that I can write $(2)$ only for integrable $J$ ? If someone could please explain what is it that I am not doing right or do not understand correctly ?...










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  • 1




    The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
    – Jason DeVito
    Nov 30 at 19:28










  • Perfect ... that answers it! you got me spot on :)
    – Kong
    Nov 30 at 19:32










  • Well, I wish I knew how to actually do the computation!
    – Jason DeVito
    Nov 30 at 19:35






  • 1




    Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
    – Jack Lee
    Nov 30 at 22:56
















2














The Nijenhuis tensor is defined to be:



$$(1):quad
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],
$$



for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:



$$(2):quad
J:TMrightarrow TMquad|quad J^2=-I_{TM}.
$$



The requirement is to show that given $J$ is integrable, $N_J$ is vanishing. I am having an issue with the following computation which gives a vanishing $N_J$, regardless of whether $J$ is integrable or not:



$$
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]
$$

$$
=[X,Y]+J^2XY-JYJX+JXJY-J^2YX-[JX,JY]
$$

$$
=[X,Y]+J^2[X,Y]
$$

$$
=[X,Y]-[X,Y]=0quadtext{(by (2) alone)}.
$$

I find this very strange because this makes absolutely no reference at all to $J$ being integrable or not, rather this follows solely from the definition of $J$ in $(2)$. Suggesting that if $J$ is an almost complex structure, then $N_J$ is always vanishing. Is it that I can write $(2)$ only for integrable $J$ ? If someone could please explain what is it that I am not doing right or do not understand correctly ?...










share|cite|improve this question




















  • 1




    The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
    – Jason DeVito
    Nov 30 at 19:28










  • Perfect ... that answers it! you got me spot on :)
    – Kong
    Nov 30 at 19:32










  • Well, I wish I knew how to actually do the computation!
    – Jason DeVito
    Nov 30 at 19:35






  • 1




    Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
    – Jack Lee
    Nov 30 at 22:56














2












2








2


1





The Nijenhuis tensor is defined to be:



$$(1):quad
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],
$$



for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:



$$(2):quad
J:TMrightarrow TMquad|quad J^2=-I_{TM}.
$$



The requirement is to show that given $J$ is integrable, $N_J$ is vanishing. I am having an issue with the following computation which gives a vanishing $N_J$, regardless of whether $J$ is integrable or not:



$$
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]
$$

$$
=[X,Y]+J^2XY-JYJX+JXJY-J^2YX-[JX,JY]
$$

$$
=[X,Y]+J^2[X,Y]
$$

$$
=[X,Y]-[X,Y]=0quadtext{(by (2) alone)}.
$$

I find this very strange because this makes absolutely no reference at all to $J$ being integrable or not, rather this follows solely from the definition of $J$ in $(2)$. Suggesting that if $J$ is an almost complex structure, then $N_J$ is always vanishing. Is it that I can write $(2)$ only for integrable $J$ ? If someone could please explain what is it that I am not doing right or do not understand correctly ?...










share|cite|improve this question















The Nijenhuis tensor is defined to be:



$$(1):quad
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],
$$



for vector fields $X$ and $Y$ on the manifold $M$, equipped with Almost Complex Structure $J$:



$$(2):quad
J:TMrightarrow TMquad|quad J^2=-I_{TM}.
$$



The requirement is to show that given $J$ is integrable, $N_J$ is vanishing. I am having an issue with the following computation which gives a vanishing $N_J$, regardless of whether $J$ is integrable or not:



$$
N_J(X,Y)equiv[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]
$$

$$
=[X,Y]+J^2XY-JYJX+JXJY-J^2YX-[JX,JY]
$$

$$
=[X,Y]+J^2[X,Y]
$$

$$
=[X,Y]-[X,Y]=0quadtext{(by (2) alone)}.
$$

I find this very strange because this makes absolutely no reference at all to $J$ being integrable or not, rather this follows solely from the definition of $J$ in $(2)$. Suggesting that if $J$ is an almost complex structure, then $N_J$ is always vanishing. Is it that I can write $(2)$ only for integrable $J$ ? If someone could please explain what is it that I am not doing right or do not understand correctly ?...







differential-geometry complex-manifolds almost-complex






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share|cite|improve this question













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edited Nov 30 at 17:11

























asked Nov 30 at 17:06









Kong

315




315








  • 1




    The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
    – Jason DeVito
    Nov 30 at 19:28










  • Perfect ... that answers it! you got me spot on :)
    – Kong
    Nov 30 at 19:32










  • Well, I wish I knew how to actually do the computation!
    – Jason DeVito
    Nov 30 at 19:35






  • 1




    Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
    – Jack Lee
    Nov 30 at 22:56














  • 1




    The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
    – Jason DeVito
    Nov 30 at 19:28










  • Perfect ... that answers it! you got me spot on :)
    – Kong
    Nov 30 at 19:32










  • Well, I wish I knew how to actually do the computation!
    – Jason DeVito
    Nov 30 at 19:35






  • 1




    Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
    – Jack Lee
    Nov 30 at 22:56








1




1




The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
– Jason DeVito
Nov 30 at 19:28




The condition $J^2 = - I$ is definitely not true just for integrable $J$. For example, $S^6$ has an almost complex structure $J$ coming from thinking of it as the unit octonians, but this almost complex structure is not integrable. In your computation, it's not clear to me that $J[JX,Y] = J^2XY - JYJX$, and similarly for the third term. I agree that $J(V+W) = JV + JW$ for any vectors $V$ and $W$, but when you write $[JX,Y] = JXY - YJX$, you have to remember that the terms $JXY$ and $YJX$ are not actually vector fields themselves, so there is no reason for $J$ to distribute over them....
– Jason DeVito
Nov 30 at 19:28












Perfect ... that answers it! you got me spot on :)
– Kong
Nov 30 at 19:32




Perfect ... that answers it! you got me spot on :)
– Kong
Nov 30 at 19:32












Well, I wish I knew how to actually do the computation!
– Jason DeVito
Nov 30 at 19:35




Well, I wish I knew how to actually do the computation!
– Jason DeVito
Nov 30 at 19:35




1




1




Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
– Jack Lee
Nov 30 at 22:56




Hint: If $X$ is any tangent vector, then $X-iJX$ is an $i$-eigenvector of $J$, and $X+iJX$ is a $(-i)$-eigenvector.
– Jack Lee
Nov 30 at 22:56















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