Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb
  R^{n+k}$   For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singular
point at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axial
curvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained using
the curvature locus (the image by the second fundamental form of the unitary
tangent vectors) and are therefore second order invariants. In fact, in the
case $n=2$ they generalise all second order curvatures which have been defined
for frontal type surfaces. We relate these curvatures with the principal
curvatures in certain normal directions of an associated regular
$(n-1)$-manifold contained in $M^n_{\mbox{sing}}$. We obtain many interesting
geometrical interpretations in the cases $n=2,3$. For instance, for frontal
type 3-manifolds with 2-dimensional singular set, the Gaussian curvature of the
singular set can be expressed in terms of the axial curvatures. Similarly for
the curvature of the singular set when it is 1-dimensional. Finally, we show
that all the umbilic curvatures which have been defined for singular manifolds
up to now can be seen as the absolute value of one of our axial curvatures.

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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th