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For singular $n$-manifolds in $\mathbb R^{n+k}$ with a corank 1 singularpoint at $p\in M^n_{\mbox{sing}}$ we define up to $l(n-1)$ different axialcurvatures at $p$, where $l=\min\{n,k+1\}$. These curvatures are obtained usingthe curvature locus (the image by the second fundamental form of the unitarytangent vectors) and are therefore second order invariants. In fact, in thecase $n=2$ they generalise all second order curvatures which have been definedfor frontal type surfaces. We relate these curvatures with the principalcurvatures in certain normal directions of an associated regular$(n-1)$-manifold contained in $M^n_{\mbox{sing}}$. We obtain many interestinggeometrical interpretations in the cases $n=2,3$. For instance, for frontaltype 3-manifolds with 2-dimensional singular set, the Gaussian curvature of thesingular set can be expressed in terms of the axial curvatures. Similarly forthe curvature of the singular set when it is 1-dimensional. Finally, we showthat all the umbilic curvatures which have been defined for singular manifoldsup to now can be seen as the absolute value of one of our axial curvatures.

Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb R^{n+k}$