On the horospherical ridges of submanifolds of codimension 2 in Hyperbolic \(n\)-space
The authors introduce the notion of horospherical ridges of submanifolds of codimention 2 in hyperbolic \(n\)-space. They prove: Theorem 1: Let \(M\) be an \(n-2\)-submanifold of hyperbolic space \(H^n_+(-1)\). Then (1) if \(n\) is odd, then \(M\) admits at least a horobinormal direction and at most \(n-2\) at each of its points. (2) If \(n\) is even and \(M\) admits a locally supporting hyperhorosphere at some points, then there is a non empty open submanifold in \(M\), all of whose points admit at least one horobinormal direction and at most \(n-2\) of them. They prove the following lemma: Let \(M\) be a regular \(n-2\)-submanifold of \(H^n_+(-1)\) and \(\nu\) a horobinormal for \(M\) at a noninflection point \(x(u)\). Then the osculating hyperhorosphere \(HS^{n-1}_\nu\) at \(x(u)\in M\) has contact of order at least 2 with the hyperhoroasymptotic line associated to \(\nu\) passing through \(x(u)\). Using the above lemma they prove Theorem 2: Under the condition of the above lemma: (1) \(x(u(0)) =\beta(0)\) is a horospherical ridge point if and only if the osculating hyperhorosphere \(HS^{n-1}_\nu\) at \(x(u(0))\in M\) has contact order at least 3 with the horoasymptotic line \(\beta\). (2) \(x(0) = \beta(0)\in M\) is a horospherical ridge point of order \(k\) if and only if the osculating hyperhorosphere \(HS^{n-1}_\nu\) has contact of order at least \(k\) with the horoasymptotic line \(\beta\).
scientific article; zbMATH DE number 2133668