[Paper Review] A_2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space

Kentaro Saji, Masaaki Umehara+1 more|arXiv (Cornell University)|Nov 6, 2010

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper establishes a realization theorem for wave fronts in space forms using coherent tangent bundles and front bundles, enabling an intrinsic formulation analogous to the fundamental theorem of surface theory. It proves that for hypersurfaces with non-negative sectional curvature in Euclidean space, $A_2$-singularities imply that the second fundamental form vanishes along the singular set if curvature is bounded, and singular principal curvatures are non-positive if curvature is non-negative.

ABSTRACT

In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space.

Motivation & Objective

• To develop an intrinsic formulation of wave fronts in space forms using front bundles and coherent tangent bundles.
• To establish a realization theorem for wave fronts in space forms, generalizing the fundamental theorem of surface theory.
• To analyze the behavior of principal singular curvatures at $A_2$-singularities in hypersurfaces with non-negative sectional curvature in $\mathbb{R}^{m+1}$.
• To provide necessary and sufficient conditions for realizing coherent tangent bundles and front bundles as smooth maps into space forms.

Proposed method

• Introduces the concept of front bundles and coherent tangent bundles as intrinsic structures for wave fronts.
• Uses a metric connection $D$ and a bundle homomorphism $\varphi$ satisfying a compatibility condition $D_X\varphi(Y) - D_Y\varphi(X) - \varphi([X,Y]) = 0$.
• Defines the $\varphi$-metric as the pullback of the inner product via $\varphi$, which is positive semidefinite.
• Applies the Gauss equation to relate the extrinsic curvature of the wave front to the intrinsic geometry of the coherent tangent bundle.
• Analyzes the limit of the external sectional curvature $K^{\text{ext}}$ near $A_2$-singularities using directional derivatives of inner products of $\varphi_i$ and $\psi_j$.
• Uses the sign of $\left\langle{D_1\varphi_1},{D_m\varphi_m}\right\rangle$ to determine the sign of singular principal curvatures.

Experimental results

Research questions

• RQ1 Under what conditions can a given front bundle be realized as a wave front in a space form of constant curvature?
• RQ2 How do the singular principal curvatures behave at $A_2$-singularities when the induced sectional curvature is bounded or non-negative?
• RQ3 What is the relationship between the extrinsic curvature of a wave front and the intrinsic geometry of its coherent tangent bundle near singularities?
• RQ4 Can the fundamental theorem of surface theory be generalized to wave fronts in space forms using intrinsic structures?
• RQ5 What constraints does non-negative sectional curvature impose on the second fundamental form along $A_2$-singularities?

Key findings

• If the sectional curvature $K$ is bounded on the regular part of a hypersurface near an $A_2$-singular set, then the second fundamental form vanishes along that singular set.
• If $K$ is non-negative on the regular part, then the singular principal curvatures along the $A_2$-singular set are all non-positive.
• The limit of the external sectional curvature $K^{\text{ext}}$ at an $A_2$-singularity is bounded below by $\delta > 0$ if and only if $\partial_m\left\langle{\varphi_1},{\psi_1}\right\rangle \cdot \partial_m\left\langle{\varphi_m},{\psi_m}\right\rangle > 0$ at the singularity.
• The sign of the singular principal curvature in the $\partial_1$-direction is determined by the sign of $\left\langle{D_1\varphi_1},{D_m\varphi_m}\right\rangle$, which is negative under non-negative curvature.
• When $K \geq \delta > 0$, the singular submanifold has positive sectional curvature.
• The Gauss equation links the sectional curvature of the singular submanifold to the product of singular principal curvatures, confirming that positivity of $K$ implies positivity of the singular submanifold's curvature.