On the topology of a compact submanifold of a sphere with bounded second fundamental form.
On the topology of positively curved Bazaikin spaces
We explore some aspects of the topology of the family of 13-dimensional Bazaikin spaces. Using the computation of their homology rings, Pontryagin classes and linking forms, we show that there is only one Bazaikin space that is homotopy equivalent to a homogeneous space, i.e., the Berger space. Moreover, it is easily shown that there are only finitely many Bazaikin spaces in each homeomorphism type and that there are only finitely many positively curved ones for a given cohomology ring. In fact,...
On the Topology of Riemann Spaces of Quasi-constant Curvature
On the topology of spherically symmetric space-times
Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected...
On the torsion of linear higher order connections
For a linear r-th order connection on the tangent bundle we characterize geometrically its integrability in the sense of the theory of higher order G-structures. Our main tool is a bijection between these connections and the principal connections on the r-th order frame bundle and the comparison of the torsions under both approaches.
On the torsion of spaces with connection
On the Total Absolute Curvature of Closed Curves in Manifolds of Negative Curvature.
On the total absolute curvature of closed curves in spheres.
On the total curvature of surfaces in
On the total (non absolute) curvature of a even dimensional submanifold Xn immersed in Rn+2.
The total curvatures of the submanifolds immersed in the Euclidean space have been studied mainly by Santaló and Chern-Kuiper. In this paper we give geometrical and topological interpretation of the total (non absolute) curvatures of the even dimensional submanifolds immersed in Rn+2. This gives a generalization of two results obtained by Santaló.
On the transverse Scalar Curvature of a Compact Sasaki Manifold
We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the...
On the trigonometry of symmetric spaces.
On the twistor bundle of de Sitter space .
On the type decomposition of the second fundamental form of a Kähler submanifold
On the uniqueness of isoperimetric solutions and imbedded soap bubbles in non-compact symmetric spaces, I.
On the uniqueness of solutions of the homogeneous curvature equations
On the uniqueness of some differential invariants: , ,
On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold
This paper divides into two parts. Let (X,ω) be a compact Hermitian manifold. Firstly, if the Hermitian metric ω satisfies the assumption that for all k, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle is nef, then for any ε >...
On the volume of a unit vector field on the three-sphere.
On the volume of unit vector fields on a compact semisimple Lie group.