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Igor Khavkine
Igor Khavkine
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symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$

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Pietro Majer
Pietro Majer
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It is well known that TS^n-1 $T\mathbb{S}^{n-1}$ is diffeomorphic to M= f^{-1}(1)$M= f^{-1}(1)$ where f:C^n---->C, f=\sum_{i=1}^{n} z_{i}^{2}$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$. two questionTwo questions: 1)Is M a symplectic submanifold of C^n ~ R^2n (with the standard symplectic structure)?

  1. Is $M$ a symplectic submanifold of $\mathbb{C}^n\sim \mathbb{R}^{2n}$ (with the standard symplectic structure)?

ifIf the answer is affirmative, we can consider two symplectic structurestructures for TS^n-1$T\mathbb{S}^{n-1}$. The first is the original structure of the tangent or cotangent bundle, the second one is the pull back structure of M $M$.

2)are these structures equivalent?

  1. Are these structures equivalent?

It is well known that TS^n-1 is diffeomorphic to M= f^{-1}(1) where f:C^n---->C, f=\sum_{i=1}^{n} z_{i}^{2}. two question: 1)Is M a symplectic submanifold of C^n ~ R^2n (with the standard symplectic structure)?

if the answer is affirmative, we can consider two symplectic structure for TS^n-1. The first is the original structure of the tangent or cotangent bundle, the second one is the pull back structure of M

2)are these structures equivalent?

It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$. Two questions:

  1. Is $M$ a symplectic submanifold of $\mathbb{C}^n\sim \mathbb{R}^{2n}$ (with the standard symplectic structure)?

If the answer is affirmative, we can consider two symplectic structures for $T\mathbb{S}^{n-1}$. The first is the original structure of the tangent or cotangent bundle, the second one is the pull back structure of $M$.

  1. Are these structures equivalent?
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Ali Taghavi
Ali Taghavi
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symplectic structure of tangent bundle

It is well known that TS^n-1 is diffeomorphic to M= f^{-1}(1) where f:C^n---->C, f=\sum_{i=1}^{n} z_{i}^{2}. two question: 1)Is M a symplectic submanifold of C^n ~ R^2n (with the standard symplectic structure)?

if the answer is affirmative, we can consider two symplectic structure for TS^n-1. The first is the original structure of the tangent or cotangent bundle, the second one is the pull back structure of M

2)are these structures equivalent?