Mixing
Symplectic structure on a covering manifold
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, omega)$ be a symplectic 2n-manifold. If there exists a covering $pi: Mto N$, then $pi^*omega$ is a symplectic form on $M$. I don't have a clue how to prove this statement. Any help would be very appreciated.
symplectic-geometry
edited Jul 24 at 13:07
Javi
2,1481725
asked Jul 24 at 10:37
usr1988
153
 |Â
show 4 more comments
up vote
1
down vote
favorite
How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, omega)$ be a symplectic 2n-manifold. If there exists a covering $pi: Mto N$, then $pi^*omega$ is a symplectic form on $M$. I don't have a clue how to prove this statement. Any help would be very appreciated.
symplectic-geometry
edited Jul 24 at 13:07
Javi
2,1481725
asked Jul 24 at 10:37
usr1988
153
2
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
1
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44
 |Â
show 4 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, omega)$ be a symplectic 2n-manifold. If there exists a covering $pi: Mto N$, then $pi^*omega$ is a symplectic form on $M$. I don't have a clue how to prove this statement. Any help would be very appreciated.
symplectic-geometry
edited Jul 24 at 13:07
Javi
2,1481725
asked Jul 24 at 10:37
usr1988
153
How to show that a covering manifold of a symplectic manifold admits a symplectic structure? More precisely, let M be a $2n$-manifold and $(N, omega)$ be a symplectic 2n-manifold. If there exists a covering $pi: Mto N$, then $pi^*omega$ is a symplectic form on $M$. I don't have a clue how to prove this statement. Any help would be very appreciated.
symplectic-geometry
edited Jul 24 at 13:07
Javi
2,1481725
asked Jul 24 at 10:37
usr1988
153
edited Jul 24 at 13:07
Javi
2,1481725
edited Jul 24 at 13:07
Javi
2,1481725
edited Jul 24 at 13:07
Javi
2,1481725
2,1481725
asked Jul 24 at 10:37
usr1988
153
asked Jul 24 at 10:37
usr1988
153
asked Jul 24 at 10:37
usr1988
153
153
2
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
1
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44
 |Â
show 4 more comments
2
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
1
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44
2
2
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
1
1
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44
 |Â
show 4 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Â
draft saved
draft discarded
Â
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2861199%2fsymplectic-structure-on-a-covering-manifold%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2
What have you tried so far ? First, observe that $pi$ is a local diffeomorphism, then try to express the pullback of the differential form $omega$ on $M$ in order to check that $pi^*omega$ is closed and non-degenerate.
– Bebop
Jul 24 at 10:54
Showing $d(pi^*omega)=0$ is easy, but I do not know how to show $(pi^*omega)^n neq 0$, without explicitly specifying a local diffeomorphism.
– usr1988
Jul 24 at 12:51
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 24 at 13:02
I think that I found an answer. Let $omega_M=pi^* omega$ be a closed 2-form on M. Then we have $int_M omega_M^n=deg(pi)int_N omega^nneq 0$(assuming M is closed and orientable), since $omega$ is non-degenerate on $N$.
– usr1988
Jul 24 at 13:04
1
Ok, so in general and assuming it is clear for you that $pi$ is a local diffeomorphism, then $forall pin M, forall Xin T_pM, (pi^*omega)_p(X)=omega_pi(p)(dpi_pi(p)(X))$. Can you check now that this 2-form is non degenerate ?
– Bebop
Jul 24 at 13:44