Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Berger curvature decomposition, Weitzenb\"ock formula, and canonical
  metrics on four-manifolds   We first provide an alternative proof of the classical Weitzneb\"ock formula
for Einstein four-manifolds using Berger curvature decomposition, motivated by
which we establish a unified framework for a Weitzenb\"ock formula for a large
class of canonical metrics on four-manifolds (or a Weitzenb\"ock formula for
"Einstein metrics" on four-dimensional smooth metric measure spaces). As
applications, we classify Einstein four-manifolds of half two-nonnegative
curvature operator, which in some sense provides a characterization of
K\"ahler-Einstein metrics with positive scalar curvature on four-manifolds, we
also discuss four-manifolds of half two-nonnegative curvature operator and half
harmonic Weyl curvature.

9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
16
The categories of this paper are: stat