Jacobi set

Definition

Consider two generic Morse functions f, g: M \to \R defined on a smooth d-manifold. Let the restriction of f to the level set g^{-1}(t) for t \in \R a regular value, be called f_t: g^{-1}(t) \to \R; it is a Morse function. Then the Jacobi set J of f and g is J = cl{{x \in M \mid x \mbox{ is critical point of } f_t }} ,

Alternatively, the Jacobi set is the collection of points where the gradients of the functions align with each other or one of the gradients vanish (cite?), for some \lambda \in \R, J = {x \in M \mid \nabla{f(x)} + \lambda \nabla{g(x)} = 0 \mbox{ or } \lambda \nabla{f(x)} + \nabla{g(x)} = 0}.

Equivalently, the Jacobi set can be described as the collection of critical points of the family of functions f+ \lambda g, for some \lambda \in \R, J = {x \in M \mid x \mbox{ is a critical point of } f + \lambda g \mbox{ or } \lambda f + g}.

References

References

1. Edelsbrunner, Herbert. (2002). "Jacobi sets of multiple morse functions". Cambridge University Press.