A theorem of Łojasiewicz asserts that any relatively compact solution of a
real analytic gradient vector field has finite length. We show here a generalization of
this result for relatively compact solutions of an analytic vector field X with a smooth
invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is
a gradient. This solves some instances of R. Thom’s Gradient Conjecture. Furthermore, if
the dimension of the ambient space is three, these solutions do not oscillate (in the sense
that they cut an analytic set only finitely many times) ; this can also be applied to some
gradient vector fields.
