TheHup said:Homework Statement
1) If f:M -> N, where M is a compact and boundaryless manifold, N is a connected manifold has regular values y and z. And h:N -> N is a diffeomorphism which is smoothly isoptopic to the identity and carries y to z, then why is z a regular value of the composition h o f?
2) Also if f,g:M -> N (M,N as above) are smoothly homotopic to each either, then why does Sard's Theorem imply that there exists an element y in N which is a regular value for both f and g?
Homework Equations
The Attempt at a Solution
1) Have been trying to think it through but it's just not clicking.
2) I know Sard's theorem tells me that the set of regular values of f is dense in N. Similarly for g, however the way I've had dense sets defined to me isn't convincing me that there has to be a point that is a regular value for both f and g.
lavinia said:1) dh is an isomorphism of tangent spaces. So dim ker f is preserved by dh.
2) the regular values of f and g are not only dense but they are open. I think this is enough to prove it.
TheHup said:Thanks for the help.
2) Is the regular values being open a consequence of Sard's theorem or something else?
Thanks again.
A diffeomorphism is a smooth, one-to-one mapping between two differentiable manifolds that preserves the smoothness of the functions involved. In simpler terms, it is a function that is both smooth and invertible, meaning that it has a smooth inverse function.
In the context of diffeomorphisms, a regular value is a point in the target manifold that is mapped to by a smooth function in the source manifold and has a full rank derivative. This means that the derivative of the function at that point has the maximum possible rank, and the function is "regular" at that point.
Regular values are important in diffeomorphisms because they allow for the construction of smooth inverse functions. If a point in the target manifold is a regular value, then the inverse function theorem guarantees the existence of a local inverse function, which can be extended to a global inverse for diffeomorphisms.
Diffeomorphisms and regular values are used in many areas of mathematics, including differential geometry, topology, and dynamical systems. They are particularly useful in studying the properties of differentiable manifolds and understanding the relationships between different spaces.
One example of a diffeomorphism is the mapping between the real line and a circle, given by the function f(x) = e^ix. A regular value in this case would be any point on the circle, as the derivative of the function is always nonzero at these points. This mapping also has a smooth inverse function, given by g(z) = ln(z), making it a diffeomorphism.