Narrow Results

In this work, we study a functional involving the generalized scalar curvatures and prove a spherical sphere theorem under some pinching condition of this quantity. As an application, we define a new invariant on $3$-dimensional manifolds and use it to study the topology of manifolds.

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hⁿ(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hⁿ(c).

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold $(M^3,g)$ with constant positive scalar curvature, if a certain combination of $L^2$ norm of the Ricci curvature and $L^2$ norm of the scalar curvature is positive, then $M^3$ is diffeomorphic to a spherical space form.

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

In this work, several pinched conditions on the Laplacian and gradient of the warping function are found in consideration of warped product submanifolds structure that force to homology groups vanish with no stable currents. Also, it is proved that a warped product pointwise semi-slant submanifold $M^n$ that is compact and oriented in an odd-dimensional spheres ${𝕊}^{2(\frac{p}{2}+q)+1}$ and ${𝕊}^{2(p+q)+1}$, has no stable integral $p+1$-currents and $2p+1$-currents, respectively, and their homology groups are null, provided squared norm of the gradient for warping function satisfies some extrinsic restrictions including the Laplacian of the warping function, pointwise slant functions in addition to dimension of fiber of warped product immersions. Moreover, under assumption of extrinsic condition on the warping function, it is show $M^n$ being homeomorphic to a standard sphere ${𝕊}^n$ with $n=4$ and homotopic to a standard sphere ${𝕊}^n$ with $n=3$. Further, the same results are generalized for contact CR-warped product submanifolds of same ambient spaces.

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

We have shown [3] that if the projective developing map of a regular curve in the sphere is injective then the curve has no self-intersection. In this paper we will show that among compact rank one symmetric spaces the only spheres have this injectivity property.