Smooth manifold and constant map

In summary, if M and N are smooth manifolds with M connected, and F:M->N is a smooth map with a zero pushforward at every point in M, then F is a constant map. This can be shown by using the fact that only constant functions can be continuous from a connected space to {0,1}, and by proving that a vanishing pushforward implies that F is locally constant. This can be done by defining a path between arbitrary points of M, which reduces the domain to [0,1], and using the theorem that connected manifolds are path connected.
  • #1
huyichen
29
0
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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  • #2
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
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.
 

1. What is a smooth manifold?

A smooth manifold is a mathematical concept used in geometry and topology to describe a space that is locally similar to Euclidean space. It is a topological space that is locally homeomorphic to a real vector space. In simpler terms, it is a space that can be smoothly and continuously mapped onto a flat space, such as a plane or a sphere.

2. What is the difference between a smooth manifold and a general manifold?

A smooth manifold is a specific type of manifold that has a set of smooth transition functions between its charts (local coordinate systems). This means that it is a differentiable manifold, and its tangent spaces are all smooth vector spaces. A general manifold, on the other hand, does not necessarily have smooth transition functions and can be more abstract in nature.

3. What is a constant map on a smooth manifold?

A constant map on a smooth manifold is a mapping from the manifold to a single point. In other words, all points on the manifold are mapped to the same point. This can also be thought of as a map that preserves the manifold's structure, as all points are mapped to the same point regardless of their position or properties.

4. How is a smooth map defined on a smooth manifold?

A smooth map on a smooth manifold is a function that preserves the smoothness of the manifold. In other words, if the manifold is locally homeomorphic to a Euclidean space, then the smooth map must preserve this local structure by mapping points to points in a smooth and continuous manner. This is important in applications such as differential geometry and physics.

5. What are some real-world applications of smooth manifolds and constant maps?

Smooth manifolds and constant maps have many applications in mathematics, physics, and engineering. Some examples include using them to describe the shape of the universe in cosmology, modeling physical systems in classical mechanics, and analyzing data in machine learning. They are also used in computer graphics to create smooth and realistic surfaces, and in robotics for motion planning and control.

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