Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

On complete submanifolds with parallel normalized mean curvature in product spaces

“…We should notice that submanifolds with constant function correspond to a natural extension of hypersurfaces with constant angle in a product space, which was widely studied by Dillen and many other authors (see, for instance [12, 13, 23, 24]).…”

Constant angle null hypersurfaces

“…In the case of semi-Riemannian spaceforms, when viewed as hyperquadrics immersed in a semi-Euclidean space, CC vector fields arise as projections of parallel vector fields defined on the respective semi-Euclidean space [34,21]. Notice that according to Eq.…”

“…Recall that in the semi-Riemannian setting, constant angle hypersurfaces can be defined via the squared norm of V /|V |, or equivalently, the squared norm of V ⊥ /|V |; where V and V ⊥ denote the tangent and normal components of V . Thus Lemma 4.2 can be interpreted as a generalization to the null context of this classical result (see for instance Lemma 3.3 in [34]).…”

“…Several classification results for constant angle semi-Riemannian (nondegenerate) hypersurfaces have been established recently in ambient spaces such as cartesian products [10,13,16,19], warped products [15,22], spaceforms [20,28,30,34] and other geometrically relevant spaces [1,36,37]. One remarkable relation between the distinguished vector field V and the geometry of a constant angle hypersurface M is encapsulated in the notion of canonical principal direction: the tangent component V ∈ Γ(T M ) is a principal direction of M [11,21,23,29].…”

Sharp Estimates for the First Stability Eigenvalue of Surfaces in the Presence of a Closed Conformal Vector Field

“…Furthermore, any Riemannian manifold possessing a non-trivial closed and conformal vector field is locally a warped product with base a real interval, and the complete ones are either the Euclidean space or the sphere with a rotationally invariant metric or a quotient of a warped product (see [12]). The presence of a closed conformal vector field in the ambient space is a very helpful property that has been considered in several works to establish classification results in recent years (see, for instance, [5,6,8,11,13]).…”