Questions tagged [co-tangent-space]
Use this tag for questions about the dual space at a point of the tangent space.
0
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0answers
13 views
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
$$ (\Lambda,\phi)^{-1}(\alpha_\Lambda,(\phi,v))= T^*L_{(\...
2
votes
2answers
32 views
Chart Transformation in $T^*_x M$
I am confused about a statement in my differential geometry script. It states:
$(U,\varphi = (x_1,...,x_n))$ and $(U,\psi = (y_1,...,y_n))$ are charts of a smooth manifold, then $$(dy_i)_x=\sum^n_{\...
2
votes
1answer
85 views
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, $df:M\rightarrow T^*M$ ...
2
votes
1answer
65 views
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 $...
1
vote
0answers
50 views
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 ...
1
vote
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 ...
2
votes
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 ∧\...
8
votes
2answers
1k views
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 ...
