On trajectories of analytic gradient vector fields on analytic manifolds

Aleksandra Nowel, Zbigniew Szafraniec

DOI: http://dx.doi.org/10.12775/TMNA.2005.008

Abstract


Let $f\colon M\to {\mathbb R}$ be an analytic proper function defined in
a neighbourhood of a closed ``regular'' (for instance semi-analytic or
sub-analytic) set $P\subset f^{-1}(y)$.
We show that the set of non-trivial trajectories of the equation $\dot x
=\nabla f(x)$ attracted by $P$ has the same Čech-Alexander
cohomology groups as $\Omega\cap\{f< y\}$, where $\Omega$ is an
appropriately choosen neighbourhood of $P$. There are also
given necessary conditions for existence of a trajectory joining two
closed ``regular'' subsets of $M$.

Keywords


Singularities; gradient vector fields

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