Singularities of caustics of surfaces in non-flat Riemannian 4-space form$^\dag$

Abstract

We propose a way to study the caustics of surfaces in non-flat Riemannian 4-space form from the viewpoint of singularity theory in this paper. As an application of the theory of Lagrangian singularity, we study the contact of surfaces with the families of hyperspheres, which is measured by the singularities of functions defined on the surfaces.

Generating family

In this section we briefly review the theories of Lagrangian singularities and generating family due to Arnol’d [1] (please also refer to [16]). Suppose that $\pi : T^*\mathbb R^r\rightarrow {\mathbb {R}}^r$ is the cotangent bundle over ${\mathbb {R}}^r$. Then the canonical symplectic structure on $T^*{\mathbb {R}}^r$ is given by the canonical 2-form $\omega =\sum _{i=1}^rdp_i\wedge du_i$, where $(u, p)=(u_1,\cdots , u_r, p_1.\cdots , p_r)$ is the canonical local coordinate on $T^*{\mathbb {R}}^r$. Let $i: L\rightarrow T^*{\mathbb {R}}^r$ be an immersion. We call i the Lagrangian immersion if $\textrm{dim}L=r$ and $i^*\omega =0$. In this case the critical value of $\pi \circ i$ is called the caustic of i and denoted by $C_L$. Let $F:({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))\rightarrow ({\mathbb {R}}, 0)$ be a family of function germs. We denote

$$\begin{aligned} C(F)=\{(x, u)\in ({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))|\frac{\partial F}{\partial x_1}(x, u)=\cdots =\frac{\partial F}{\partial x_n}(x, u)=0\} \end{aligned}$$

and call it the catastrophe set of F. We also call

$$\begin{aligned} {\mathcal {B}}(F)=\{u\in ({\mathbb {R}}^r, 0)|~\mathrm{there~ exist~} (x, u)\in C(F)~ \mathrm{such~ that}~ {\textrm{rank}}\left( \frac{\partial ^2F}{\partial x_i\partial x_j}(x, u)\right) < n\} \end{aligned}$$

the bifurcation set of F. Let $\pi _r:({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))\rightarrow ({\mathbb {R}}^r, 0)$ be the canonical projection, then we can show that the bifurcation set of F is the critical value set of $\pi _r|_{C(F)}$.

We call F is a Morse family if the map germ

$$\begin{aligned} \Delta F=\left( \frac{\partial F}{\partial u_1},\cdots ,\frac{\partial F}{\partial u_r}\right) :({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))\rightarrow ({\mathbb {R}}^r, 0) \end{aligned}$$

is non-singular, where $(x, u)=(x_1, \cdots , x_n, u_1, \cdots , u_r)\in ({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))$. In this case we have a smooth submanifold germ $C(F)\subset ({\mathbb {R}}^n\times {\mathbb {R}}^r, (0,0))$, and then we can define a map germ $L(F): (C(F), 0)\rightarrow T^*{\mathbb {R}}^r$ by

$$\begin{aligned} L(F)(x, u)=\left( u, \frac{\partial F}{\partial u_1},\cdots ,\frac{\partial F}{\partial u_r}\right) . \end{aligned}$$

We can easily show that L(F) is a Lagrangian immersion. Then we have the following well known theorem [1].

Theorem A.1

All Lagrangian submanifold germs in $T^*{\mathbb {R}}^r$ are constructed by the above method. F is called a generating family of L(F).