Smooth manifold and constant map

  • Thread starter huyichen
  • Start date
  • #1
29
0

Main Question or Discussion Point

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 useful in solving the problem?
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,773
8
Show that if M is connected, then a locally constant map F:M-->N is (every pt has an open nbhd where F is constant) is constant.

Then show that a vanishing pushfoward implies that F is locally constant.
 
  • #3
236
0
If you define a path between arbitrary points of M, your domain becomes [0,1] which is easier to work with. If you're using Lee (I think I remember this exercise) you have as a theorem that connected manifolds are path connected.
 

Related Threads for: Smooth manifold and constant map

Replies
1
Views
2K
Replies
1
Views
4K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
2
Replies
27
Views
5K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
Top