Extrinsic geometric properties of shortest geodesics in a neighborhood about a point of strict tangency (Q1903850)

The present paper is a continuation of the author's previous one [All-Union Conference of Geometry, Kishinëv, 252-253 (1988)]. Let \(M\) be an \(m\)-dimensional surface lying in the \(n\)-dimensional Euclidean space \(E^n\), \(1\leq m< n\), \(n\geq 3\). On a surface \(M\) the concept of a point of strict tangency is formulated using the language of limit sets. The aim of the work is to prove that a point of strict tangency of \(M\) is also a point of strict tangency for each of the shortest geodesics passing through the point and regarded as curves in \(E^n\). Theorem 1: If a shortest geodesic \(\gamma\) on an \(m\)-dimensional surface \(M\) in \(E^n\) passes through a point \(X\) of strict tangency, then it has a tangent at this point, and \(X\) is a point of strict tangency of \(\gamma\). Theorem 2: Let \(X\) be a point of strict tangency of \(M\). In every neighborhood \(V\subset M\) about \(X\) there is a neighborhood \(W\subset M\) about \(X\) such that two arbitrary points of \(W\) can be joined by a shortest geodesic passing through \(V\).