Smooth manifold and constant map

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SUMMARY

The discussion establishes that if M and N are smooth manifolds with M connected, and F: M -> N is a smooth map whose pushforward is the zero map at each point in M, then F must be a constant map. The reasoning is based on the property that continuous functions from connected spaces to discrete spaces, such as {0,1}, are constant. Additionally, it is shown that a vanishing pushforward implies that F is locally constant, leading to the conclusion that F is indeed constant due to the path-connected nature of connected manifolds.

PREREQUISITES
  • Understanding of smooth manifolds and their properties
  • Familiarity with the concept of pushforward in differential geometry
  • Knowledge of continuous functions and their behavior on connected spaces
  • Basic principles of topology, particularly related to locally constant functions
NEXT STEPS
  • Study the properties of smooth manifolds in differential geometry
  • Learn about the pushforward and its implications in smooth maps
  • Explore the relationship between connectedness and path-connectedness in topology
  • Investigate locally constant functions and their applications in manifold theory
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to deepen their understanding of smooth manifolds and their mappings.

huyichen
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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?
 
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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.
 
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.
 

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