Recent content by robforsub

That's the part that is confusing to me. So I have checked on wikipedia, and it defines E*=Hom(E,MXR). However, there is a natural isomorphism on bundle that is Hom(E,E')=E*(direct sum)E, therefore I am wondering if I can use this isomorphism to get the result that E* is isomorphic to E*(direct...

As E* is defined in some book as Hom(E, MXR). What could be the isomorphism between dual vector bundle E* and Hom(E, MXR)?

If given a n*n matrix with all rows and columns sum to 0, how do I argue that all its (n-1)*(n-1) minor have the same determinant up to a sign? Since all rows and columns all sum to 0, then I know that any column is a linear combination of all others, so that the determinant of this n*n matrix...

Let me clarify it a little bit, so I want to show that the image of that curve y is a immersed submanifold of manifold M, and without the condition y'(t)!=0 for all t, I can not say y is a immersion, then the image of curve y is immersed submanifold, right?

Is a periodic curve still an immersed submanifold of a manifold M? Suppose y is the curve map an interval to a manifold M, and y is periodic, which means it is not injective. And immersed submanifold must be the image of a injective immersion.

Is every orientation form w on a compact smooth manifold closed?(i.e. dw=0)

Thanks for clarification on smooth homotopy part, then with smooth homotopy established, we just use Stokes's theorem, since any n-form on N representing orientation will be closed, since N is compact Hausdorff space, then d\Omega_N=0, which gives integral of F^*\Omega_N on M is equal to...

But I just don't see what's the point of using Whitney approximation theorem in this prob.

I think here the homotopy should be smooth as well, since both F and G are diffeomorphisms.

orientation preserving is defined as if for each p in M, F_* takes the oriented bases of TpM to oriented bases of TF(p)N And I don't think it requires to use homology related concepts to solve this problem

This is actually a problem from Lee's Introduction to smooth manifolds 14-21: Let M and N be compact, connected, oriented, smooth manifolds. and suppose F, G:M->N are diffeomorphisms. If F and G are homotopic, show that they are either both orientation-preserving or both...

What if there is a smooth function F:C^2\{0} to C, defined as F(z1,z2)=z1^p+z2^q with p and q>=2 and they are relatively prime, then how to show that S^3 intersect F^(-1)(0) is diffeomorphic to 2-tori?

My mistake, it should be how 2-tori is embedded into S^3

Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3

Let M1,..,Mk be smooth manifolds, and let Pi_j be the projection from M1XM2X...XMk->Mj. Show that the map a:T_(p1,...,pk)(M1XM2X...Mk)->T_p1(M1)\oplus...\oplusT_pk(Mk) a(X)=(Pi_1*X,Pi_2*X,...,Pi_k*X) is an isomorphism. The way I am thinking to prove the statement is to show that a is a...