Recent content by huyichen

It is a well fact that tensor product is associative up to isomorphism, but how should I use Universal property(you know, diagrams that commute) to show that it is true?

M is a smooth manifolds, and X is a vector field on M, y is a maximal integral curve of X. Now suppose y is periodic and nonconstant, show that there exists a unique positive number T(called the period of y) such that y(t)=y(t') if and only if t-t'=kT for some integer k.(For this problem, What...

But as F- and F+ agrees on equator, then the limit indeed hold, right?

Actually, this is problem 14-6 from Introduction to Smooth Manifolds, Lee's book, we can not directly talk about the boundary because F+, F- is continuous but not smooth up to the equator. And the integrand is unbounded, but if we interpret in an appropriate limiting sense, then we can show...

Justify means that we can indeed replace Integral F^*w with the limit integral as R--> 1

Define F+:B2->S2 by F+=(u,v)=(u,v,sqrt(1-u^2-v^2)) and F-:B2->S2 by F-=(u,v)=(u,v,-sqrt(1-u^2-v^2)) Then to show that Integral of w on S2=Integral of F+^*(w) on B2-Integral of F-^*(w), why do we need to justify the limits(As the integral on the right hand side are defined as limits as...

For a direct sum of Mobius band, it is trivial if it has two linear independent nonvanishing sections. I have the following as my sections： s1=(E^(i*theta), (Cos(theta/2), Sin(theta/2)) s2=(E^(i*theta), (-Sin(theta/2), Cos(theta/2)) Clearly, the above sections are linearly independent and...

We can show that any vector field V:M->TM(tangent bundle of M) is smooth embedding of M, but how do we show that these smooth embeddings are all smoothly homotopic? How to construct such a homotopy?

If K and L are embedded manifold of M, and T_p(K intersect L)=T_p K intersect T_p L and K intersect L is again a embedded manifold , then we say K intersect L is clean

How to show that a transverse intersection is clean, but not conversely?

"if a vector field has only nondegenerate zeros then the number of zeros is bounded" With no idea how to show that without using Poincare-Hopf Theorem. Any proof possible without using any concept from algebraic topology?

Now F:S^2->R^4 is a map of the following form: F(x,y)=(x^2-y^2,xy,xz,yz) now using the smooth covering map p:S^2->RP^2, p is the composition of inclusion map i:S^2->R^3 and the quotient map q:R^3\{0}->RP^2. show that F descends to a smooth embedding of RP^2 into R^4. Is the problem asked to...

Show the intersection of complex sphere (|z1|^2+|z2|^2+|z3|^2=1) in C^3 and the complex cone (z1^2+z2^2+z3^3=1) in C^3 is a smooth submanifold of C^3. I am trying to do it using regular level set, but I am not sure which one of (1,0) or (1,1,0) should be set to be the regular value?

But, don't we need to take component functions of arbitrary X into account?

Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be...