Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tangent cones and $C^1$ regularity of definable sets   Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an
o-minimal structure. We prove that the following three statements are
equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the
paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in
X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear
subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in
$x$, and the density $\theta(X, x) < 3/2$.

12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin