Pythagorean theorem \& curvature with lower or upper bound (Q2133973)

Let \(\left( M,\rho\right) \) be a geodesically connected metric space. A point \(x\in M\) is said to be of CBB type (resp. CBA type) if there is a neighborhood \(U_{x}\) of \(x\) and \(k_{x}\in\mathbb{R}\) such that Alexandrov's curvature at every point of \(U_{x}\) is at least \(k_{x}\) (at most \(k_{x}\)). The authors say that \(\left( M,\rho\right) \) is an Alexandrov space if it is locally compact and if every point of \(\left( M,\rho\right) \) is of CBB type (resp. CBA type). The authors introduce the following condition for \(k\in\mathbb{R}\) and \(x\in M\):\newline(P\(_{k}\left( x\right) \)) There is a neighborhood \(V_{x}\) of \(x\), such that for every \(q\in V_{x}\) and every geodesic segment \(\gamma_{r_{1},r_{2}}\) in \(V_{x}\) not containing \(q\), if there is \(p\) strictly inside \(\gamma_{r_{1},r_{2}}\), such that \(\rho\left( p,q\right) =\min_{s\text{ in }\gamma_{r_{1},r_{2}}}\rho\left( q,s\right) \), then \[ \widetilde{\angle}_{k}qpr_{1},\widetilde{\angle}_{k}qpr_{2}\leq\frac{\pi} {2}(\text{resp. }\geq\frac{\pi}{2}). \] The authors' main result (Theorem A) states that if \(x\) is a point of a complete Alexandrov space \(\left( M,\rho\right) \) (if \(x\) is a point of CBB type, then the authors require that \(x\) has a finite dimensional neighborhood in \(M\)), then \(\left( M,\rho\right) \) is of curvature at least \(k\) (at most \(k\)) around \(x\) if and only if condition \(P_{k}\left( x\right) \) holds. In addition, the authors prove the rigidity in Theorem A when \(\widetilde{\angle}_{k}qpr_{1},\widetilde{\angle}_{k}qpr_{2}=\frac{\pi}{2}\).