[Paper Review] Singularities of tangent surfaces to generic space curves

Goo Ishikawa, Tatsuya Yamashita|arXiv (Cornell University)|Feb 8, 2016

Geometric Analysis and Curvature Flows17 references3 citations

TL;DR

This paper classifies the generic singularities of tangent surfaces to space curves in arbitrary dimension and affine connection geometry. It generalizes classical results from Euclidean 3-space to general manifolds with torsion-free affine connections, proving that for generic curves, the only singularities are cuspidal edges (in 3D) or embedded cuspidal edges (in higher dimensions), with precise criteria based on higher-order covariant derivatives of the curve.

ABSTRACT

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities.

Motivation & Objective

• To extend the classification of singularities of tangent surfaces from Euclidean 3-space to general Riemannian or affine manifolds of arbitrary dimension.
• To resolve the gap in the literature by treating non-projectively flat geometries, which had not been systematically analyzed before.
• To provide intrinsic, differential-geometric criteria for identifying cuspidal edge and folded umbrella singularities using covariant derivatives of the curve.
• To establish a genericity condition in the Whitney $C^\infty$ topology for curves, ensuring that only the specified singularities appear.
• To explore the emergence of more degenerate singularities (e.g., $(2,5)$-cuspidal edge) in non-generic cases, particularly for torsionless curves in non-flat spaces.

Proposed method

• Define the $\nabla$-tangent surface as the ruled surface generated by $\nabla$-geodesics tangent to a curve $\gamma$ in a manifold $M$ with affine connection $\nabla$.
• Use the covariant derivatives $\nabla\gamma$, $\nabla^2\gamma$, $\nabla^3\gamma$, and $\nabla^4\gamma$ to characterize the local type of singularity at a point $t_0$.
• Apply singularity theory criteria for frontals and map-germs, particularly focusing on the rank of the Jacobian and the behavior of the characteristic function.
• Employ the notion of 'openings' from previous work to generalize results from projective geometry to general affine connections.
• Use local diffeomorphism classification via the condition that two map-germs are equivalent if they are related by diffeomorphisms of the source and target.
• Prove the main theorems by reducing the problem to known normal forms (e.g., cuspidal edge: $(t,s) \mapsto (t+s, t^2 + 2st, t^3 + 3st^2)$) and verifying equivalence under the given geometric structure.

Experimental results

Research questions

• RQ1 What are the generic singularities that can appear on the $\nabla$-tangent surface to a generic curve in a manifold of dimension $m \geq 3$ with a torsion-free affine connection?
• RQ2 How do the classical singularities (cuspidal edge, folded umbrella) in 3D Euclidean space generalize to higher-dimensional and non-flat ambient spaces?
• RQ3 Under what conditions on the covariant derivatives of the curve does the $\nabla$-tangent surface exhibit a cuspidal edge or folded umbrella singularity?
• RQ4 Can more degenerate singularities, such as the $(2,5)$-cuspidal edge, appear on $\nabla$-tangent surfaces, and if so, under what geometric conditions?
• RQ5 What is the role of torsionlessness and projective flatness in determining the type of singularity on the tangent surface?

Key findings

• For $\dim(M) = 3$, the $\nabla$-tangent surface to a generic curve has only cuspidal edge or folded umbrella singularities, depending on the linear independence of $\nabla\gamma$, $\nabla^2\gamma$, $\nabla^3\gamma$.
• For $\dim(M) \geq 4$, the only generic singularities are embedded cuspidal edges, with the same linear independence condition on the first three covariant derivatives.
• The folded umbrella singularity occurs precisely when $\nabla\gamma$, $\nabla^2\gamma$, $\nabla^3\gamma$ are linearly dependent but $\nabla\gamma$, $\nabla^2\gamma$, $\nabla^4\gamma$ are linearly independent.
• In non-projectively flat spaces, torsionless curves can give rise to non-fold singularities such as the $(2,5)$-cuspidal edge, which is not possible in flat or projectively flat geometries.
• The $\nabla$-tangent surface to a torsionless curve in a locally projectively flat manifold is always a fold singularity, as shown via projective equivalence to a planar curve in Euclidean space.
• The paper provides a complete local diffeomorphism classification of singularities in the $C^\infty$ Whitney topology, establishing a genericity condition for the curve space.