Submanifolds with constant scalar curvature

Hypersurfaces with Constant Scalar Curvature

Let M be a complete two-dimensional surface immersed into the three-dimensional Euclidean space. Then a classical theorem of Hilbert says that when the curvature of M is a non-zero constant, M must be the sphere. On the other hand, when the curvature of M is zero, a theorem of Har tman-Nirenberg [4] says that M must be a plane or a cylinder. These two theorems complete the classification of com...

Constant scalar curvature metrics with isolated singularities

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by...

Slant submanifolds with prescribed scalar curvature into cosymplectic space form

In this paper, we have proved that locally there exist infinitely many three dimensional slant submanifolds with prescribed scalar curvature into cosymplectic space form M 5 (c) with c ∈ {−4, 4}while there does not exist flat minimal proper slant surface in M 5 (c) with c 6= 0. In section 5, we have established an inequality between mean curvature and sectional curvature of the subamnifold and ...

Constant scalar curvature metrics on toric surfaces

Euler Characters and Submanifolds of Constant Positive Curvature

This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if Mn is an n-dimensional compact connected Riemannian submanifold of constant positive curvature in E2n−1, then Mn must be simply connected. It also gives a conformal version of this theorem.