Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mean curvature, volume and properness of isometric immersions   We explore the relation among volume, curvature and properness of a
$m$-dimensional isometric immersion in a Riemannian manifold. We show that,
when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq
p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded
geometry, properness is equivalent to the finiteness of the volume of extrinsic
balls. We also relate the total absolute curvature of a surface isometrically
immersed in a Riemannian manifold with its properness. Finally, we relate the
curvature and the topology of a complete and non-compact $2$-Riemannian
manifold $M$ with non-positive Gaussian curvature and finite topology, using
the study of the focal points of the transverse Jacobi fields to a geodesic ray
in $M$ . In particular, we have explored the relation between the minimal focal
distance of a geodesic ray and the total curvature of an end containing that
geodesic ray.

30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat