Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

Abstract

We define cuspidal curvature $\kappa _c$ (resp. normalized cuspidal curvature $\mu _c$) along cuspidal edges (resp. at a swallowtail singularity) in Riemannian 3-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $\kappa _\varPi ^{}$ called the product curvature (resp. $\mu _\varPi ^{}$ called normalized product curvature) of $\kappa _c$ (resp. $\mu _c$) and the limiting normal curvature $\kappa _\nu$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $\kappa _\varPi ^{}$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.

The first author was partly supported by CAPES and JSPS under Brazil-Japan research cooperative program, Proc BEX 12998/12-5. The second author was partly supported by Grant-in-Aid for Scientific Research (C) No. 26400087 from the Japan Society for the Promotion of Science, the third author by (A) No. 26247005 and the fourth author by (C) No. 26400087 from the Japan Society for the Promotion of Science.

Appendix: The Coordinate Invariance of Blow up

We give here the procedure of blowing up and show its coordinate invariance, which is used to define rational boundedness and continuity in Definition 3.4.

We define the equivalence relation $\sim$ on $\varvec{R}\times S^1$ by

$$ (r,\theta )\sim (-r,\theta +\pi ), $$

where $S^1:=\varvec{R}/2\pi \mathbf Z$. We set ${\mathcal M}:=\varvec{R}\times S^1/\sim$, namely, ${\mathcal M}$ is the quotient space of $\varvec{R}\times S^1$ by this equivalence relation. We also denote by

$$ \pi :\varvec{R}\times S^1\rightarrow {\mathcal M}$$

the canonical projection. Let $(\varvec{R}^2;u,v)$ be the (u, v)-plane. Then there exists a unique $C^\infty$-map $\varPhi :{\mathcal M}\rightarrow \varvec{R}^2$ such that

$$ \varPhi \circ \pi (r,\theta ):=(r\cos \theta ,r\sin \theta ) \qquad ((r,\theta )\in \varvec{R}\times S^1). $$

This map $\varPhi$ gives the usual blow up of $\varvec{R}^2$ at the origin.

From now on, we show that the coordinate invariance of this blow up procedure: let $(\varvec{R}^2;U,V)$ be the (U, V)-plane, and consider a diffeomorphism $f:(\varvec{R}^2;u,v)\rightarrow (\varvec{R}^2;U,V)$ such that $f(0,0)=(0,0)$. Then we can write

$$ f\circ \varPhi \circ \pi (r,\theta ) =(U(r,\theta ),V(r,\theta )). $$

Since $f(0,0)=(0,0)$, it holds that $U(0,\theta )=V(0,\theta )=0$. Then the well-known division property of $C^\infty$-functions yields that there exist $C^\infty$-function germs $\xi (r,\theta )$ and $\eta (r,\theta )$ such that

$$ U(r,\theta )=r \xi (r,\theta ),\qquad V(r,\theta )=r \eta (r,\theta ). $$

Since f is a diffeomorphism, one can easily show that $\xi (0,\theta )^2+\eta (0,\theta )^2$ is positive for all $\theta \in S^1$, and the $C^\infty$-function

$$ R(r,\theta ):=r \sqrt{\xi (r,\theta )^2+\eta (r,\theta )^2} $$

is defined on $\varOmega :=(f\circ \varPhi \circ \pi )^{-1}\bigl (\{(U,V)\,;\, U^2+V^2<\varepsilon ^2\}\bigr )$ for sufficiently small $\varepsilon >0$. Moreover, there exists a unique $C^\infty$-function $\varTheta :\varOmega \rightarrow S^1$ such that

$$ \cos \varTheta (r,\theta )=\frac{\xi (r,\theta )}{ \sqrt{\xi (r,\theta )^2+\eta (r,\theta )^2}}, \quad \sin \varTheta (r,\theta )=\frac{\eta (r,\theta )}{ \sqrt{\xi (r,\theta )^2+\eta (r,\theta )^2}}. $$

Then, the $C^\infty$-map $F:\pi (\varOmega )\rightarrow \varPhi ^{-1} (\{(U,V)\,;\,U^2+V^2<\varepsilon ^2\})$ satisfying the property $F\circ \pi (r,\theta )=\pi (R(r,\theta ),\varTheta (r,\theta ))$ is uniquely determined, and satisfies the relation $F\circ \varPhi =\varPhi \circ f$. By our construction, such a map F depends only on f. Hence, by replacing f by $f^{-1}$, we can conclude that F is a diffeomorphism for sufficiently small $\varepsilon >0$.

Let p be a point on a 2-manifold $\varSigma ^2$. The above construction of F implies that we can define the ‘blow up’ of the manifold $\varSigma ^2$ at p.