Spherical Submanifolds of a Euclidean Space

A1 and A2 being arbitrary constants. A natural generalization of this question to higher dimensions could be: ‘given an isometric immersion ψ : Mn → Rn+2 of a compact n-dimensional Riemannian manifold (Mn, g), obtain conditions for ψ(Mn) ⊂ Sn+1(c), where Sn+1(c) is the sphere of constant curvature c’. We write ψT , ψ⊥ as tangential and normal components of the position vector ψ in Rn+p and show that for a connected submanifold ψ : Mn → Rn+p, ψ(Mn) ⊂ Sn+p−1(c) for some c if and only if 〈H, ψ⊥〉 is a constant, where H is the mean curvature vector field (Theorem 4.1). Similarly it is observed that for this submanifold the vector field ψT being harmonic also provides a necessary and sufficient condition for ψ(Mn) ⊂ Sn+p−1(c), for some constant c (Theorem 4.3). We also study codimension-2 isometric immersions ψ : Mn → Rn+2 of a compact connected Riemannian manifold (Mn, g) with parallel mean curvature vector field and observe that if the sectional curvatures of Mn are strictly positive and the scalar curvature S of Mn satisfies S < n(n−1)λ−2, then no such isometric immersion of (Mn, g) is contained in a ball of radius λ in Rn+2 (Theorem 5.1).