Mixing
![Does this series converges to some constant? [closed]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZCTB4-bb-s-32SAm6ytjzFOU06cH9o8q-a_wq_UnH6lcd8hvws23CmBWHM6g256LzTX9yK1EiCCzTC1NSgtxlNk_J9hdr9bA3tD0Nqntbl521j4bLAm_uXPANuWLBhcCKTKzns2gaUodx/s72-c/1.jpg)
Vertical $J$-holomorphic spheres
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Let $pi:Mto N$ 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 $$TMcong TFoplus pi^astTN,$$ where $TF$ is the tangent bundle along the fiber.
Suppose $F=S^2$ and we can define an almost complex structure $J$ on $M$ by $$J=J_S^2oplus J_pi^astTN,$$ where $J_S^2$ is the standard complex structure on the fiber $S^2$ and $J_pi^astTNin mathrmEnd(pi^astTN)$ satisfies $J_pi^astTN^2=1$. Then by construction, a fiber of $pi$ is a $J$-holomorphic sphere in $(M,J)$.
Now suppose that $tildepi:Mto N$ is another $S^2$-bundle which is isomorphic to $pi$. Then $tildepi$ gives rise to another splitting $$TMcong TF;tildeoplus ;tildepi^astTN.$$ Further, suppose we define another almost compelx structure $tildeJ$ on $M$ $$tildeJ=J_S^2;tildeoplus ;J_tildepi^astTN$$ as above. As above, a fiber of $tildepi$ is a $tildeJ$-holomorphic sphere.
Questions: How are $J$ and $tildeJ$ related? More specifically, is a fiber of $tildepi$ also a $J$-holomorphic sphere (and vice-versa)?
differential-geometry symplectic-geometry almost-complex
asked Jul 27 at 1:28
rpf
931512
add a comment |Â
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Let $pi:Mto N$ 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 $$TMcong TFoplus pi^astTN,$$ where $TF$ is the tangent bundle along the fiber.
Suppose $F=S^2$ and we can define an almost complex structure $J$ on $M$ by $$J=J_S^2oplus J_pi^astTN,$$ where $J_S^2$ is the standard complex structure on the fiber $S^2$ and $J_pi^astTNin mathrmEnd(pi^astTN)$ satisfies $J_pi^astTN^2=1$. Then by construction, a fiber of $pi$ is a $J$-holomorphic sphere in $(M,J)$.
Now suppose that $tildepi:Mto N$ is another $S^2$-bundle which is isomorphic to $pi$. Then $tildepi$ gives rise to another splitting $$TMcong TF;tildeoplus ;tildepi^astTN.$$ Further, suppose we define another almost compelx structure $tildeJ$ on $M$ $$tildeJ=J_S^2;tildeoplus ;J_tildepi^astTN$$ as above. As above, a fiber of $tildepi$ is a $tildeJ$-holomorphic sphere.
Questions: How are $J$ and $tildeJ$ related? More specifically, is a fiber of $tildepi$ also a $J$-holomorphic sphere (and vice-versa)?
differential-geometry symplectic-geometry almost-complex
asked Jul 27 at 1:28
rpf
931512
Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37
add a comment |Â
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up vote
3
down vote
favorite
Let $pi:Mto N$ 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 $$TMcong TFoplus pi^astTN,$$ where $TF$ is the tangent bundle along the fiber.
Suppose $F=S^2$ and we can define an almost complex structure $J$ on $M$ by $$J=J_S^2oplus J_pi^astTN,$$ where $J_S^2$ is the standard complex structure on the fiber $S^2$ and $J_pi^astTNin mathrmEnd(pi^astTN)$ satisfies $J_pi^astTN^2=1$. Then by construction, a fiber of $pi$ is a $J$-holomorphic sphere in $(M,J)$.
Now suppose that $tildepi:Mto N$ is another $S^2$-bundle which is isomorphic to $pi$. Then $tildepi$ gives rise to another splitting $$TMcong TF;tildeoplus ;tildepi^astTN.$$ Further, suppose we define another almost compelx structure $tildeJ$ on $M$ $$tildeJ=J_S^2;tildeoplus ;J_tildepi^astTN$$ as above. As above, a fiber of $tildepi$ is a $tildeJ$-holomorphic sphere.
Questions: How are $J$ and $tildeJ$ related? More specifically, is a fiber of $tildepi$ also a $J$-holomorphic sphere (and vice-versa)?
differential-geometry symplectic-geometry almost-complex
asked Jul 27 at 1:28
rpf
931512
Let $pi:Mto N$ 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 $$TMcong TFoplus pi^astTN,$$ where $TF$ is the tangent bundle along the fiber.
Suppose $F=S^2$ and we can define an almost complex structure $J$ on $M$ by $$J=J_S^2oplus J_pi^astTN,$$ where $J_S^2$ is the standard complex structure on the fiber $S^2$ and $J_pi^astTNin mathrmEnd(pi^astTN)$ satisfies $J_pi^astTN^2=1$. Then by construction, a fiber of $pi$ is a $J$-holomorphic sphere in $(M,J)$.
Now suppose that $tildepi:Mto N$ is another $S^2$-bundle which is isomorphic to $pi$. Then $tildepi$ gives rise to another splitting $$TMcong TF;tildeoplus ;tildepi^astTN.$$ Further, suppose we define another almost compelx structure $tildeJ$ on $M$ $$tildeJ=J_S^2;tildeoplus ;J_tildepi^astTN$$ as above. As above, a fiber of $tildepi$ is a $tildeJ$-holomorphic sphere.
Questions: How are $J$ and $tildeJ$ related? More specifically, is a fiber of $tildepi$ also a $J$-holomorphic sphere (and vice-versa)?
differential-geometry symplectic-geometry almost-complex
asked Jul 27 at 1:28
rpf
931512
edited Jul 27 at 1:34
asked Jul 27 at 1:28
rpf
931512
asked Jul 27 at 1:28
rpf
931512
asked Jul 27 at 1:28
rpf
931512
931512
Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37
add a comment |Â
Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37
Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37
Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37
add a comment |Â
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Without some distinguished parametrization of every fibers, how can you say that $J_S^2$ is the standard complex structure on the fibers? Similarly and in a related way, you established that the fibers are images of $J$-(holomorphic maps from the )sphere and of $tildeJ$-sphere, but there might not be for the same maps from the sphere. Just think of the trivial case $N = pt$.
– Jordan Payette
Jul 28 at 12:37