Gradient-like vector field

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In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient 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.

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Definition

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.

Formally:¹

away from critical points, X ⋅ f > 0 , {\displaystyle X\cdot f>0,}
around every critical point there is a neighborhood on which f is given as in the Morse lemmas:

f ( x ) = f ( b ) − x 1 2 − ⋯ − x α 2 + x α + 1 2 + ⋯ + x n 2 {\displaystyle f(x)=f(b)-x_{1}^{2}-\cdots -x_{\alpha }^{2}+x_{\alpha +1}^{2}+\cdots +x_{n}^{2}}

and on which X equals the gradient of f.

Dynamical system

The associated dynamical system of a gradient-like vector field, a gradient-like dynamical system, is a special case of a Morse–Smale system.

An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: Gradient-like vector fields, p. 56–69
Gradient-Like Vector Fields Exist Archived 2012-03-26 at the Wayback Machine, September 25, 2009

References

p. 63 https://books.google.com/books?id=TtKyqozvgIwC&pg=PA63 ↩