2
votes
2answers
56 views

Using a non-zero wedge product to write a set of vectors as a linear combination of another set of vectors in a finite dimensional space.

Question: Let $V$ be a finite dimensional vector space, and let $ \{ v_1, ..., v_r\}$ and $\{w_1, ..., w_r\}$ be two sets of vectors in $V$. Suppose that $\sum_{i=1}^{r} v_i \wedge w_i = 0$, and ...
0
votes
1answer
34 views

Vanishing of a covector (1-form) and a vector field

a) A one-form $\theta$ is zero if and only if $\theta X = 0$ for all $X$ in the set $\frak{X} \mathrm{(M)}$ of all smooth vector fields on a manifold $M$. b) A vector field $X$ is zero if and only ...
1
vote
1answer
23 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
1
vote
1answer
79 views

On tangent spaces of Steifel Manifolds

I was trying to read Edelman et al.'s 1998 paper "The Geometry of Algorithms with Orthogonality Constraints" and since I don't have any differential geometry or much linear algebra background I am ...
2
votes
1answer
47 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why folloing sequence is exact? $0\to \mathbb{R} \to C^\infty (M)\to A\to 0$ Which $A$ here is the set of Global Hamiltonian vector fields.
6
votes
1answer
107 views

Tangential Space of a differentiable manifold is always $\mathbb R^n$?

Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties ...
0
votes
1answer
106 views

Problem related to differential of a map

I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
1
vote
2answers
136 views

The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...
1
vote
0answers
75 views

transformation of symplectic structure by a matrix

Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have $$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$ Let ...
1
vote
0answers
343 views

Is linear manifold a pure algebraic concept?

I was wondering if linear manifold is a pure algebraic concept? Here is its definition from planetmath: Suppose $V$ is a vector space and suppose that $L$ is a non-empty subset of $V$. If there ...