# Questions tagged [co-tangent-space]

Use this tag for questions about the dual space at a point of the tangent space.

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### Explicit evaluation of symplectic form on cotangent bundle

Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle. Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$. Let $p\in U$ and $v\in T^*_pM$, then we can write $v = v_i dx^i$ ...
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### First Chern class of the cotangent bundle vanishes

I'm interested in the first Chern class of the cotangent bundle. I concretely work on the sphere $S^2$, but the reasoning below seems to work for any manifold. I take the symplectic point of view ...
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### Problem with goniometric function and triangles.

I can't resolve this problem, above there is what I've done. Can you help me? Is there a better way to solve this problem? A triangle $\triangle{ABC}$ is given, where the angle $\angle{BAC}=2{\alpha}$ ...
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### References on the ''bi-symplectic'' structure of $T^\ast T \mathcal M$.

Let $\mathcal M$ be a smooth manifold. One can consider two tautological one forms on $T^\ast (T \mathcal M)$: ("Diagonal case") The usual tautological one form on $T^\ast \mathcal E$, ...
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### How do we construct the cotangent space?

I am following a graduate course in algebraic geometry and our professor introduced last week the cotangent space at a point p of a variety as the quotient $m/m^2$ where $m$ is the maximal ideal of ...
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### Different notation and a little more insight for $dx^i$ based on an alternative definition of the cotangent space

In this post the cotangent space $T_p^* M$ was defined as the quotient $C^\infty(M)/W_p(M)$ where $W_p(M) \subset C^\infty(M)$ is the subspace of smooth functions with vanishing 1st derivative (i.e. ...
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### Metric, tangent and contangent spaces

I have a lot of confusion understanding manifolds, charts, tangent and cotangent vector spaces… Please let me know where I am wrong. Let’s take a flat minkosky space in 4 dimensions and call it $M$, ...
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### Possible definition of the cotangent space

This definition is inspired by the book mentioned in this question. As it turns out later it is identical to that in question mentioned in the comment by Paul Frost. The tangent space at $p \in M$ can ...
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### Contangent space as a jet space, (inconsistency ?), Renteln

In Renteln's, Manifolds, Tensors and Forms, p. 81, The cotangent space as a jet space$^*$, we have the following definitions Let $f:M \to \mathbb R$ be a smooth function, $p \in M$, and $\{x^i\}$ ...
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### Confusion about role of covectors in behavior of momentum

This question has been beaten to death on math stackexchange, so I'm a little embarassed I have to ask again, but I am really confused about how/why momentum is a covector. I read Arnold but that didn'...
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### symplectic geometry: help showing the cotangent lift of an action to a symplectic manifold is a symplectic action

I am following da Silva's lectures on symplectic geometry. She defines the lift of a diffeomorphism as follows: Let $X_1$ and $X_2$ be $n$-dimensional manifolds with cotangent bundles $M_1=T^*X_1$ and ...
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### What are differential forms?

For a manifold $M$, if we want to speak of "tangent vectors," we often say the tangent bundle $TM$ is the space of tangent vectors. This is sort of an abuse of terminology, I guess you could say, ...
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### Struck in Diffeomorphism invariance of the line integral

I have been trying to study The Cotangent bundle from An introduction to smooth manifolds by John M Lee. I have been struck at a specific point in the Diffeomorphism Invariance of the integral. The ...
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### Higgs bundle and spectral curve

I have trouble understanding the following part in E. Wittens paper "More On Gauge Theory And Geometric Langlands": The context is the following: $(E, \varphi)$ is a Higgs bundle with structure group ...
In the article 'More on Gauge Theory and Geometric landlands' by Edward Witten, I have read the term of a 'Canonical bundle'. I've looked that up and as far as I understand, the definition is : $M$ ...
I have some questions about definitions from Otto Forster's Lectures on Riemann Surfaces. The setup: Let $X$ be a Riemann surface, and allow $U \subset X$ to be open. Define $\mathcal{E}(U)$ to be ...