Real Algebraic and Analytic Geometry |

Relative Poincare Lemma, Contractibility, Quasi-Homogeneity and Vector Fields Tangent to a Singular Variety.

e-mail: , ,

Submission: 2004, January 12.

*Abstract:
We study the interplay between the
properties of the germ of a singular variety $N\subset \Bbb R^n$
given in the title and the algebra of vector fields tangent to
$N$. The Poincare lemma property means that any closed
differential $(p+1)$-form vanishing at any point of $N$ is a
differential of a $p$-form which also vanishes at any point of
$N$. In particular, we show that the classical quasi-homogeneity
is not a necessary condition for the Poincare lemma property, it
can be replaced by quasi-homogeneity with respect to a smooth
submanifold of $\Bbb R^n$ or a chain of smooth submanifolds. We
prove that $N$ is quasi-homogeneous if and only if there exists a
vector field $V, V(0)=0,$ which is tangent to $N$ and has positive
eigenvalues. We also generalize this theorem to quasi-homogeneity
with respect to a smooth submanifold of $\Bbb R^n$.*

Mathematics Subject Classification (2000): 32B10, 14F40, 58K50.

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