Real Algebraic and Analytic Geometry
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62. Aleksandra Nowel, Zbigniew Szafraniec:
On trajectories of analytic gradient vector fields on analytic manifolds.

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Submission: 2003, October 16.

Abstract:
Let $f:M\longrightarrow {\R}$ 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 \v{C}ech--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 $M$.

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