Extrinsic geometry and higher order contacts of surfaces in R^5

Journal of Singularities
volume 17 (2018), 193-213

Received: 8 July 2017. Received in revised form: 19 May 2018.

DOI: 10.5427/jsing.2018.17h

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Abstract:

We study the extrinsic geometry of a surface in R^5 in relation to contact theory. We first completely determine the numerical invariants of the second fundamental form and describe the corresponding curvature ellipse. We then introduce and study a new quadratic map closely related to the degenerate directions of the surface, we characterize inflection and umbilic points of the surface in terms of the invariants, and we obtain an intrinsic equation of the asymptotic lines. Finally, we give a simple condition which guarantees the existence of an isometric reduction of codimension of the surface into R^4

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