in this paper‎, ‎the‎ ‎authors prove that a strictly kähler-berwald manifold with‎ ‎nonzero constant holomorphic sectional curvature must be a‎ kähler manifold‎.

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on S2 × S2 the authors of [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on S2 × R3): ”The boundary S2 × S2 of the S2 × B3 ⊂ S2 × R3 with an arbitrary complete metric of non-negative sectional curvature contains a poin...

For an n-dimensional complete connected Riemannian manifold M with sectional curvature KM ≥ 1 and diameter diam(M) > π2 , and a closed connected totally geodesic submanifold N of M , if there exist points x ∈ N and y ∈ M satisfying the distance d(x, y) > π 2 , then N is homeomorphic to a sphere. We also give a counterexample in 2-dimensional case to the following problem: let M be an ndimension...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...

The Riemannian product $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ , where $${\mathbb{M}}_i(c_i)$$ denotes the 2-dimensional space form of constant sectional curvature $$c_i \in {\mathbb{R}}$$ has two different $${\mathrm{Spin}^{\mathrm{c}}}$$ structures carrying each a parallel spinor. restriction these spinor fields to 3-dimensional hypersurface M characterizes isometric immersion into...

In this article, the totally geodesic submanifolds in the complex 2Grassmannian G2(C) and in the quaternionic 2-Grassmannian G2(H) are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2 published by Chen and Nagano (1978) is incomplete. For example, G2(H) with n ≥ 5 contains totally...

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-CartanVranceanu family. This gives a positive answer to a conjecture formulated in [2]. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Ga...

Abstract We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of operators, among which Pucci’s extremal some singular operators such as those modeled the p - $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Ma...

We show that the unit tangent bundle of S4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature. AMS Classi cation numbers Primary: 53C20 Secondary: 53C20, 58B20, 58G30