Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Riemannian manifolds with positive weighted Ricci curvature of
  negative effective dimension   In this paper, we investigate complete Riemannian manifolds satisfying the
lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for
the negative effective dimension $N<0$. We analyze two $1$-dimensional examples
of constant curvature $\mathrm{Ric}_N \equiv K$ with finite and infinite total
volumes. We also discuss when the first nonzero eigenvalue of the Laplacian
takes its minimum under the same condition $\mathrm{Ric}_N \ge K>0$, as a
counterpart to the classical Obata rigidity theorem. Our main theorem shows
that, if $N<-1$ and the minimum is attained, then the manifold splits off the
real line as a warped product of hyperbolic nature.

10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th