Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Common zeros of inward vector fields on surfaces   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 Poincar\'e-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.

26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc