In the present paper, we revisit the rigidity of hypersurfaces in Euclidean space. We highlight Darboux equation and give new proof of rigidity of hypersurfaces by energy method and maximal principle.
For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of
Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones.
We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.
We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions. On the othe
r hand, involutes of framed curves are direct generalizations of involutes of regular space curves and frontals in the Euclidean plane. We investigate properties of normal surfaces, circular evolutes, and involutes of framed curves. We can observe that taking circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similarly to evolutes and involutes of fronts in the Euclidean plane. Furthermore, we investigate the relations among singularities of normal surfaces, circular evolutes, and involutes of framed curves.
In this paper we give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case our assumptions are only of intrinsic nature.
In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.