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i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the...

@mathwonk yes, and in the embedded submanifold {z=2} the the tangent vectors are just in the form a∂∂x+b∂∂y, so the two values of two forms are the same.

@WWGD , thank you very much, so I think I am right, doesn't it?

I think the tangent vectors of the plane z=2 are in the form of a$\frac{\partial}{\partial x}$+b$\frac{\partial}{\partial y}$, to a$\frac{\partial}{\partial x}$+b$\frac{\partial}{\partial y}$, both of the two forms of the value, am I right?

Thank you, so what should I do?

Assume M=xdy -ydx+dz ∈ Ω1(R^3). What's the restriction of M to the plane {z=2}? I think it's xdy-ydx. Is that right?

1 Let A → N, B → N be two vector bundles over a manifold N. How to show that there is a vector bundle Hom(A, B) whose fiber above x ∈ N is Hom(A, B)x := Hom(Ax, Bx)? 2 Let A → N, B → N be two vector bundles over a manifold N. Let C∞(A, B) denote the space of maps of vector bundles from A to B...

Dear Lavinia, thank you！

Suppose x ∈ Ω^(n−1)(Rn \{0}) is closed and the integral of x on S^(n-1) equals to 1. I am stuck on how to show there does not exist an n − 1 form y ∈ Ω(n−1)(R^n) with y|R^n\{0} = x.