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Geometry of submanifolds. I: The first Casorati curvature indicatrices. (English) Zbl 1473.53041

From the text: In this series of papers, we will present some geometrical properties and formula’s concerning what we propose to be called the Trenčevski frame on submanifolds with arbitrary dimensions and of arbitrary co-dimensions in (semi-)Riemannian ambient spaces. In the present part, attention will be confined to \(n\)-dimensional submanifolds \(M\) of co-dimension \(m\) in \((n+m)\)-dimensional Euclidean spaces. The Trenčevski frames may well be the adapted frames on general submanifolds of which the geometrical characterisations of the tangent and normal orthonormal vectors are as natural as possible, being essentially involved with our kind’s most intuitive notions of curvature. Hereafter, as such, only the tangent and the first principal normal vector fields will be discussed, the hereby involved curvatures being the tangential and the first normal Casorati curvatures, which, in case of surfaces \(M^2\) in \(E^3\) are nothing but the squares of the Euler normal or principal curvatures and Casorati’s “most common sense” curvature, respectively.

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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