Questions tagged [cotangent-bundles]

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34 votes
6 answers
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Kähler structure on cotangent bundle?

The total space of cotangent bundle of any manifold $M$ is a symplectic manifold. Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure? Please support your claim with reference or ...
Mohammad Farajzadeh-Tehrani's user avatar
9 votes
2 answers
4k views

Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems). A ...
Pengfei's user avatar
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6 votes
2 answers
1k views

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

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 ...
Ali Taghavi's user avatar
5 votes
1 answer
954 views

Does the preimage of the Slodowy slice in $T^*G/P$ have a name?

Let $G$ be your favorite simple complex Lie group, and $P\subset G$ your favorite parabolic subgroup. We can identify $T^*G/P$ with the space of pairs $$\{(gP,x)\in G/P\times \mathfrak g | x\perp \...
Ben Webster's user avatar
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20 votes
1 answer
2k views

Functoriality of the cotangent bundle

Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle ...
Theo Johnson-Freyd's user avatar