|3||Clarification in response to a question in the comments.|
Edit: This is a response to Andrew's question below (since answering in the comments proves difficult).
I'm using the word "Morse" loosely, largely because I don't actually know Morse theory. I would suggest that the definition I give is better than what's traditionally used. What I actually mean is this:
Let M be a smooth manifold and f:M→R a smooth map. What type of object is the second derivative f(2)? In general, you should not talk about it by itself, because it does not transform as a tensor, although the pair (f(1),f(2)) is a vector in the 2-jet bundle over M. But if c is a critical point of f, then f(2)(c) is naturally a symmetric bilinear form TcM x TcM → R. Thus it is a map TcM→T*cM. All I ask is that this map have zero kernel.
But if this condition is too weak, whereas a reasonable stronger condition works, I'd love to hear it.
Critical points on a fiber bundle
Consider a (smooth) bundle E→_B_, and a (smooth) function f: E → R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting of all points for which all fiber-wise derivatives of f vanish, so that upon intersecting with any fiber C consists of the critical points of the restriction of f to the fiber. If the fiber is n-dimensional, then C is carved out by n equations, and so generically has codimension n in E.
Let's say that c is a point in C so that the second derivative of f in the fiber is nondenegerate (i.e. has non-degenerate Hessian; i.e. f restricts to a Morse function on the fiber through c). Does it follow that the projection C→_B_ is a local diffeomorphism near c?
The answer is yes when everything is finite-dimensional (and I believe the statement is iff). I am interested in the case when B is finite-dimensional but the fibers of E are infinite-dimensional.