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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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Cotangent lift of left translation SE(3)

I am wondering how I can verify the cotangent lift of left translation $ T^*L_{(\Lambda,\phi)}:T^*_{(\Lambda,\phi)}G\to T^*_eG$ which reads $$ (\Lambda,\phi)^{-1}(\alpha_\Lambda,(\phi,v))= T^*L_{(\...
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Chart Transformation in $T^*_x M$

I am confused about a statement in my differential geometry script. It states: $(U,\varphi = (x_1,...,x_n))$ and $(U,\psi = (y_1,...,y_n))$ are charts of a smooth manifold, then $$(dy_i)_x=\sum^n_{\...
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1answer
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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, $df:M\rightarrow T^*M$ ...
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1answer
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If $S\subseteq M$ is a submanifold, is there a canonical way to identify $T_{p}^{*}S$ as a subspace of $T_{p}^{*}M$?

I have a few questions. Any thoughts to any one of them will be appreciated. Suppose $S\subseteq M$ is an embedded submanifold of $M$. There is a convenient characterization of the tangent spaces of $...
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0answers
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Construction of group structure in T^{*}G where G is a Lie group

Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51: "The group $G$ acts naturally on itself by left translations, as well as by ...
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1answer
66 views

Cotangent lift of the commutator of 2 vector fields

Given a smooth manifold $X$ and a vector field $u$ on $X$, we can perform a cotangent lift to get a vector field $\tilde u$ on $T^*X$. Explicitly, this can be done by thinking of $u$ as defining an ...
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1answer
121 views

Wedge product of a 2-form with a 1-form.

(Wedge product of a 2-form with a 1-form).* Let $\omega$ be a $2-$form and $\tau$ a $1-$ form on $\mathbb R^3$. If $X, Y, Z$ are vector fields on $M$, find an explicit formula for $(\omega ∧\...
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Why is the momentum a covector?

Can someone tell me why the momentum is an element of the cotangent space? More detailed: if we have some smooth manifold M and the cotangent space $T_{x}M^{*}$ I know that the momentum p is an ...