Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Bochner Technique and Weighted Curvatures   In this note we study the Bochner formula on smooth metric measure spaces. We
introduce weighted curvature conditions that imply vanishing of all Betti
numbers.

6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph