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Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups. (English) Zbl 1203.43007
A Carnot group is a connected, simply connected graded nilpotent Lie group equipped with some Carnot-Carathéodory metric. The authors introduce a notion of convexity of hypersurfaces in Carnot groups in terms of the horizontal second fundamental form of their graphs. The concept of a horizontal second fundamental form was proposed by R. K. Hladky and S. D. Pauls [J. Differ. Geom. 79, No. 1, 111–139 (2008; Zbl 1156.53038)]. The main result characterizes smooth convex functions on a Carnot group in terms of positive semidefiniteness of the horizontal second fundamental form of their graph utilizing a Riemannian approximation scheme.

MSC:
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
35H20 Subelliptic equations
52A41 Convex functions and convex programs in convex geometry
53A35 Non-Euclidean differential geometry
53C17 Sub-Riemannian geometry
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