Common zeros of inward vector fields on surfaces
Abstract
A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\Phi^X$. A compact relatively open set K in the zero set of X is a block. The Poincaré-Hopf index is generalized to an index for blocks that may meet the boundary. A block with nonzero index is essential. Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ and let K be an essential block of zeros for X. Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $\Phi^Y$ preserves area. Applications are made to actions of Lie algebras and groups.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2012
- DOI:
- 10.48550/arXiv.1204.1301
- arXiv:
- arXiv:1204.1301
- Bibcode:
- 2012arXiv1204.1301H
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Group Theory;
- Mathematics - Geometric Topology;
- 54H15;
- 54H25