Geodesic space, orthogonality between geodesics and nonexistence of local points in Hadamard spaces
Let \((X, d)\) be a metric space. Following mainly Alexandrov and Busemann, the author recalls first the notion of geodesics or triangles on \(X\), and injective radius and then defines the notion of the orthogonality between geodesics on \(X\). Next, the author defines for a triangle on \(X\), the notion of \(\text{CAT}_k\) triangle for \(k\in \mathbb{R}\) and the notion of curvature bounded by \(k\), denoted by \(K_X\leq k\). The space \(X\) is called a geodesic space if for any two points \(p\) and \(q\in X\) there exists a minimal geodesic connecting \(p\) and \(q\). The author proves that if \(X\) is a simply connected locally compact complete geodesic space with injectivety radius \(\delta\) then for any \(\delta\)-minimal complete geodesic \(\sigma: \mathbb{R}\to X\) and any point \(p\not\in\sigma(\mathbb{R})\) with \(d(p,\sigma(\mathbb{R}))< \delta/2\), there exists a geodesic \(\sigma_1\) through \(p\), which is orthogonal to \(\sigma\). Finally, the author defines the notion of a focal point, for a geodesic on \(X\) and proves that if \(X\) is a complete geodesic space with \(K_X\leq 0\) then the universal covering of \(X\) has no focal point.
scientific article; zbMATH DE number 1933100