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.