Taut submanifolds

Taut Submanifolds and Foliations

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb Z_2$-taut. We explicitely construct generalized Bott-Samelson cycles for the critical points of the energy functionals on the path spaces of a taut submanifold which, generically, represent a basis for the $\mathbb Z_2$-cohomology. We also consider singular Riemannian foliations all of whose leaves are taut. Using our…

Taut 与 Dupin 子流形（更新版）

Taut and Dupin Submanifolds (Updated Version)

本文更新了1997年关于欧氏空间中taut与Dupin子流形的综述，补充了后续研究成果，详述等参超曲面及两者关系，并给出主要分类结果。

Taut submanifolds are algebraic

We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s.

光滑Dupin超曲面的解析性与紧致性

Analyticity and tautness of smooth Dupin hypersurfaces

本文证明满足Dupin条件的光滑浸入欧氏超曲面局部实解析且半代数，并由此推出紧致嵌入超曲面具有紧致性。

接触阶与直纹子流形

Order of contact and ruled submanifolds

本研究借助初等几何测度论，推广了关于密切性与直纹子流形的Monge-Cayley-Salmon定理。

An elementary approach to tangent space variation on Riemannian submanifolds

We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidean geometry. We also show that using the tangent variation result one can prove a new structural property of local feature size function. This structural property is a generalization of a result of Giesen and Wagner [GW04, Lem. 7].