Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rigidity theorems for submetries in positive curvature   We derive general structure and rigidity theorems for submetries $f: M \to
X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$.
When applied to a non-trivial Riemannian submersion, it follows that $diam X
\leq \pi/2 $. In case of equality, there is a Riemannian submersion $\mathbb{S}
\to M$ from a unit sphere, and as a consequence, $f$ is known up to metric
congruence. A similar rigidity theorem also holds in the general context of
Riemannian foliations.

19
The categories of this paper are: eess
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys