All Questions
Tagged with cotangent-bundles dg.differential-geometry
11 questions
8
votes
1
answer
806
views
Derivations on the continuous functions of a manifold
For a manifold $M$ a vector field is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I ...
- 969
3
votes
0
answers
141
views
Is there a precisely formulable obstruction for the tangent bundle being a Lie algebra bundle?
Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...
- 18.4k
14
votes
3
answers
2k
views
Splitting of tangent bundle
Is it possible to give an example of $n$ dimensional manifold with the property that the tangent bundle $TM$ cannot be expressed as Whitney sum of two subbundles? It is certain true for two sphere; it ...
- 9,340
6
votes
1
answer
734
views
Are the Sasaki metrics on tangent and cotangent bundle isomorphic?
Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...
- 2,481
2
votes
0
answers
643
views
A symplectic structure for cotangent bundle
Before that I mention my question explicitly, I start with my motivation:
Look at $\mathbb{D} \times \mathbb{C}=\{(x_{1},x_{2},y_{1},y_{2})\mid x_{1}^{2}+x_{2}^{2}< 1\}$.This can be identified ...
- 366
11
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 ...
- 2,244
1
vote
1
answer
493
views
Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?
Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a ...
- 285
-4
votes
1
answer
468
views
Symplectic forms and 1-forms [closed]
Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple ...
- 1,025
1
vote
0
answers
221
views
Co-normal bundle of orthogonal compliment
Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
- 2,649
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 ...
- 54.6k
34
votes
6
answers
5k
views
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 ...
