We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain newglobal rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer–Gromoll theorems by showing that any compact n -dimensional submanifold of \mathbb R^{n+p} is singularly genuinely rigid in \mathbb R^{n+q} , for any q < \mathrm {min}\{5, n\}–p . Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

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We show that a real Kähler submanifold in codimension 6 6 is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity.

We consider Riemannian n-manifolds M with nontrivial κ-nullity "distribution" of the curvature tensor R, namely, the variable rank distribution of tangent subspaces to M where R coincides with the curvature tensor of a space of constant curvature κ (κ ∈ R) is nontrivial. We obtain classification theorems under diferent additional assumptions, in terms of low nullity/conullity, controlled scalar curvature or existence of quotients of finite volume. We prove new results, but also revisit previous ones.

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor's conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.