Curves with finite turn. (English) Zbl 1167.46321

Summary: We study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by A.D.Alexandrov, A.V.Pogorelov, and Yu.Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one-sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results than those that were proved previously with geometric methods in Euclidean spaces.


46T99 Nonlinear functional analysis
53A04 Curves in Euclidean and related spaces
58B99 Infinite-dimensional manifolds
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