Recent content by JYM

Ok, Thanks.

I want to solve the following problem: Suppose B=B(0,R) be a ball in C^n, n>1. Let f be holomorphic in B and continuous on B closure. If f(a)=0 for some a in B, show that there is p in boundary of B such that f(p)=0. I assumed f(p) is non zero for every point p in boundary B and create...

No. We can construct such a sequence. Now I get the idea; the result follows from first countablity of manifolds ( as second countable is first countable).

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold. My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a...

Lavinia, A homeomorphism is not necessarly closed.

Oh, thanks, I see it. 1/f cannot be extended to all of M since f is zero at p. I see the fact that S is embedded follows from the following fact but I can't justify. Let M be a manifold and ϕ : S → M be an injective immersion. Show that ϕ is an embedding if and only if every smooth function f ...

I try to solve the following problem: If S be submanifold of M and every smooth function f on S has a smooth extentsion to all of M, then S is properly embedded. [smooth means C-infinity]. I can show that S is embedded. What I need is to show either S is closed in M or the inclusion map is...

Thanks.

Let $h$ be a bump function that is $0$ outside $B_\epsilon^m(0)$ and posetive on its interior. Let $f$ be smooth function on $B_{2\epsilon}^m(0)$. Define $f^*(x)=h(x)f(x)$ if $x\in B_{2\epsilon}^m(0)$ and $=0$ if $x\in \mathbb{R^m}-B_\epsilon^m(0)$. I want to show that $f^*$ is smooth on...

I new for the group action concept and the problem is before that. what i want to use is charts. i upload my trial.

Yes it is correct. It is given that the map F is smooth only on U.

I have been stuck several days with the following problem. Suppose M and N are smooth manifolds, U an open subset of M , and F: U → N is smooth. Show that there exists a neighborhood V of any p in U, V contained in U, such that F can be extended to a smooth mapping F*: M → N with...

I see it. Thanks!

I want to show that a closed unit ball is manifold with boundary and I attempted as uploaded. But I am not happy with the way I showed the boundary chart is injective. Am I right?