Blaschke Manifolds Research Articles

Projective Blaschke manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds.

Spherical rank rigidity and Blaschke manifolds

In this paper we give a characterization of locally compact rank one symmetric spaces, which can be seen as an analogue of Ballmann's and Burns and Spatzier's characterizations of nonpositively curved symmetric spaces of higher rank, as well as of Hamenstädt's characterization of negatively curved symmetric spaces. Namely, we show that a complete Riemannian manifold M is locally isometric to a compact, rank one symmetric space if M has sectional curvature at most 1 and each normal geodesic in M has a conjugate point at π .

Pointed Blaschke manifolds and geodesic normal sections

We study complete submanifolds of Euclidean space where every geodesic passing through a fixed point is the normal section along it. We prove that all such geodesics are independent of the direction at the point and such submanifolds are pointed Blaschke manifolds or diffeomorphic to a Euclidean space.

Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.

Smooth great circle fibrations and an application to the topological Blaschke conjecture

We study great smooth circle fibrations of round spheres and Blaschke manifolds of the homotopy type of complex projective spaces.

Unitary-symmetric Kählerian manifolds and pointed Blaschke manifolds

Unitary-symmetric Kählerian manifolds and pointed Blaschke manifolds

On Blaschke manifolds and harmonic manifolds

On Blaschke manifolds and harmonic manifolds

On almost Blaschke manifolds II

where Br(Om) denotes the metric r-ball centered at the origin o m in the tangent space TraM and ExPm denotes the exponential mapping from T~M onto M. In general the diameter d(M) is equal to or greater than the injectivity radius i(M). In special case when the diameter is equal to the injectivity radius, we say that such (M, g) is a Blaschke manifold. For example compact symmetric spaces of rank one so called CROSSes have such a property. The famous Blaschke conjecture asks; Is any Blaschke manifold isometric to a CROSS? It is known that Blaschke manifolds have the cohomology types of the CROSSes, although in I-2, 11] R.Bott and H.Nakagawa have shown that this conclusion holds under much weaker assumption. Moreover M. Berger proved that Blaschke manifolds which are diffeomorphic to spheres or real projective spaces are isometric to standard ones ([1]). Recently it was shown that any Blaschke manifold is homeomorphic to one of the CROSSes in low dimensions by H. Gluck, F. Warner and C.T. Yang [7]. See also H. Sato and T. Mizutani [14, 15]. In this paper we study the following problem from a different view point. What is the topology of a riemannian manifold whose injectivity radius is close to its diameter? We get the following results in the lower dimensional case.

Blaschke manifolds of the projective plane type

Blaschke manifolds of the projective plane type

Blaschke manifolds with taut geodesics

Blaschke manifolds with taut geodesics