1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Question about regular surfaces

  1. Jun 4, 2012 #1
    Hello, I have been trying to solve the following problem about regular surfaces from Do Carmo's book of differential geometry of curves and surfaces, section 2-3, exercise 14.

    1. The problem statement, all variables and given/known data

    Problem: Let A[itex]\subset[/itex]S be a subset of a regular surface S. Prove that A is a regular surface if and only if A is open in S; that is, A=U[itex]\cap[/itex]S, where U is an open set in R[itex]^{3}[/itex]

    A regular surface is a subset S[itex]\subset[/itex]R[itex]^{3}[/itex] such that for each p[itex]\in[/itex]S, there exists a neighborhood V in R[itex]^{3}[/itex] and a map x: U -> V[itex]\cap[/itex]S of an open set U[itex]\subset[/itex]R[itex]^{2}[/itex] onto V[itex]\cap[/itex]S such that:
    1. x is differentiable
    2. x is a homeomorphism
    3. dx[itex]_{p}[/itex]:R[itex]^{2}[/itex]->R[itex]^{3}[/itex] is one-to-one

    3. The attempt at a solution

    I have solved the implication: A open in S [itex]\Rightarrow[/itex] A regular surface, my problem is with the other part of the implication. One way to show that A is open in S, is to show that for each point in A there is an open set V of S, containing the point and such that V [itex]\subset[/itex]A, and another way is to show that A=U[itex]\cap[/itex]S, where U is and open set in R[itex]^{3}[/itex].
    If A and S are regular surfaces, with A[itex]\subset[/itex]S and p[itex]\in[/itex]A, then I can find a map x[itex]_{s}[/itex]: U->V[itex]_{s}[/itex] and a map x':U'->V[itex]_{a}[/itex] where U and U' are open sets in R[itex]^{2}[/itex] and V[itex]_{s}[/itex] and V[itex]_{a}[/itex] are open sets in S and A respectively, such that conditions 1,2 and 3 are satisfied. However, I don't know how to use this to guarantee the existence of an open set V in S such that p[itex]\in[/itex]V and V [itex]\subset[/itex]A, or and open set in R[itex]^{3}[/itex] such that A=U[itex]\cap[/itex]S.

    I would like some tips (not the solution) on how to "attack" the problem. Thanks.
    (Sorry if my english is bad).
    Last edited: Jun 4, 2012
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted