Gradient-like vector field
The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.
Given a Morse function f on a manifold M, a gradient-like vector field X for the function f is, informally:
- away from critical points, X points "in the same direction as" the gradient of f, and
- near a critical point (in the neighborhood of a critical point), it equals the gradient of f, when f is written in standard form given in the Morse lemmas.
- away from critical points,
- around every critical point there is a neighborhood on which f is given as in the Morse lemmas:
and on which X equals the gradient of f.
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