Splitting, parallel gradient and Bakry-Emery Ricci curvature
Abstract
In this paper we obtain a splitting theorem for the symmetric diffusion operator
Δ
ϕ
=Δ−⟨∇ϕ,∇⟩
and a non-constant
C
3
function
f
in a complete Riemannian manifold
M
, under the assumptions that the Ricci curvature associated with
Δ
ϕ
satisfies
Ric
ϕ
(∇f,∇f)≥0
, that
|∇f|
attains a maximum at
M
and that
Δ
ϕ
is non-decreasing along the orbits of
∇f
. The proof uses the general fact that a complete manifold
M
with a non-constant smooth function
f
with parallel gradient vector field must be a Riemannian product
M=N×R
, where
N
is any level set of
f
.