# 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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### Notation on tensors and Hamiltonians

I have been reading some notes and at point the author uses a notation I am not aware of and I wanted to figure out what it was. Let $T^*M$ be the cotangent bundle of a compact manifold $M$, and ...
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### Cotangent space as a quotient

Let $M$ be a smooth manifold, $p \in M$. Let $I_p$ be the subspace of $C^{\infty}\left ( M \right )$ consisting of smooth functions that vanish at $p$, and let $I_p^2$ be the product of $I_p$ with ...
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### Relation between different definitions of the cotangent space of a smooth manifold

There are two ways known to me to define the cotangent bundle on a smooth manifold $X$: Either as the dual bundle of the tangent bundle (see any textbook on differential geometry) or (by abuse of ...
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### 1-forms vs k-forms vs Vector Fields

I'm trying to get a comprehensive understanding of similarities and differences of differential forms and vector fields. As I understand, 1-forms are analogous to vector fields in that for a vector ...
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### Confused regarding the (dual) basis vectors of the cotangent space $T^*_pM$ of a smooth manifold $M$ and components.

Let $M$ be a smooth $n$-manifold with $p\in M$. Consider the cotangent space $$\operatorname{Alt}^{1}T_pM = \operatorname{Hom}(T_pM,\mathbb{R}) = T^*_pM$$ Now, choosing a chart $(U,\varphi)$ around $p$...
1 vote
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### Pullback of a Volume Form

Suppose we have a a diffeomorphism between two manifolds, $f: M \rightarrow N$ and a volume form $\Omega$ on $N$. Then is it true that $f^{*}(\Omega) = \Upsilon$ will always be a volume form on $M$? ...
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### Second derivatives, Hamilton and tangent bundle of tangent bundle TTM

I'm learning the Hamilton formalism of classical mechanics, where a second order differential equation is formalized as two first order differential equations on the cotangent bundle of the ...
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### Tautological 1-form on the cotangent bundle is intrinsic using transformation properties

I'm following this lecture in symplectic geometry and I'm trying to show the result stated at 31 minutes that the canonical 1-form on the cotangent bundle $M = T^*X$ is well defined regardless of ...
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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, ...
1 vote
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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 ...
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### Canonical bundle definition

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 : $M$ ...
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### Question concerning cotangent spaces of Riemann Surfaces

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 ...
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### How do we define the cotangent space as the quotient of ideals?

I am interested in the definition of the cotangent space as the quotient space of ideals. The definition goes like this: Let $\mathcal M$ be a smooth manifold. $C^\infty (\mathcal M)$ is the ring of ...
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### Hamiltonian vector field on cotangent bundle

I want to proof the following: Let $Y: Q \rightarrow TQ$ be a global vector field with corresponding flow $\psi_t$. Now let $X:T^{*}Q \rightarrow T(T^{*}Q)$ be the vector field generated by the ...
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### Smooth function $y$ that satisfies $dy|_p=w$ for covector $w$.

If $M$ is a smooth manifold, $w\in T^*_pM$, then is it possible to find a smooth function $y:M\to \mathbb{R}$ such that $dy|_p=w$. If it is, is there an easy way to give $y$ explicitly? I have just ...
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### Krull's Principal Ideal Theorem for tangent spaces

In "Foundations of Algebraic Geometry" by Ravi Vakil (page $333$, problem $12.1.B$) there is the following problem Suppose $A$ is a ring, and $m$ a maximal ideal. If $f ∈ m$, show that the Zariski ...
### Tangent space $T_q(df(M))$ as a subspace of $T_q(T^*M)$
I have been asked to describe the tangents space $T_q(df(M))$ as a subspace of $T_q(T^*M)$ where $f\in C^\infty(M)$ and $df$ is a 1-form (or smooth section of $T^*M$). Here, $df:M\rightarrow T^*M$ ...