Blaschke manifolds of the projective plane type
A closed Riemannian manifold M is called a Blaschke manifold if the conditions 1) the points p and q are conjugate, and 2) \(d(p,q)=diam M\), are equivalent. It is known that \(H^*(M,{\mathbb{Z}})=H^*(\bar M,{\mathbb{Z}})\) where \(\bar M\) is one of \((S^ n,KP^ n,Ca P^ 2)\) and \(K={\mathbb{R}},{\mathbb{C}}\) or \({\mathbb{H}}\). The author studies the case \(\bar M=KP^ 2\). He obtains the following results: for all points p and q the set cut(p)\(\cap cut(q)\) consists of one point and therefore M may be endowed with an abstract projective structure where cut(p) plays the role of a straight line (theorem 1). If \(d(p,q)=diam M\) and \(\gamma_ i\), \(i=1,2\), are geodesics joining p and q then \[ <({\dot \gamma}_ 1(p),{\dot \gamma}_ 2(p))=<({\dot \gamma}_ 1(q),{\dot \gamma}_ 2(q)) \] (theorem 2). For the case \(\bar M={\mathbb{C}}P^ n\) an almost complex structure and a closed 2-form may be constructed on the set cut(p) which is not degenerated for \(n=2\). Due to the limited space of this note only short sketches are given here instead of full proofs.
scientific article; zbMATH DE number 3944744