### On trajectories of analytic gradient vector fields on analytic manifolds

#### 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 $M$.

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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