Closed orientable surfaces and fold Gauss maps

Abstract

This paper describes how the elliptic and hyperbolic regions of a surface are related to stable Gauss maps on closed orientable surfaces immersed in three-dimensional space. We will show that for certain connected, closed, orientable surfaces containing a finite number of embedded circles that delineate two distinct types of regions, if all regions of one type are homeomorphic to a cylinder, then there exists an immersion $f: M \rightarrow \mathbb {R}^3$ for which the Gauss map is a fold Gauss map.

1 Introduction

The singularities of a stable Gauss map of a closed and orientable surface generically immersed in Euclidean 3-space, in Whitney’s sense, were described in [2]. The singular set is formed by fold curves with isolated cusp points and is called the parabolic set on the surface. Each parabolic curve that belongs to this set separates a hyperbolic region from an elliptic region of the surface.

We will demonstrate a sort of converse, for some connected closed orientable surfaces M containing a finite number of embedded circles that separate the surface into two types of regions $M_H$ and $M_E$, with each $M_H$ region being homeomorphic to a cylinder, then there exists an immersion $f: M \rightarrow \mathbb {R}^3$ associated to a stable Gauss map without cusps.

2 Stable Gauss maps

Let M and P be smooth surfaces and $f,\,g: M\rightarrow P$ be smooth maps between them. The maps f and g are $\mathcal {A}$-equivalent (or equivalent) if there are orientation-preserving diffeomorphisms $l: M\rightarrow M$ and $k: P\rightarrow P$, such that $g \circ l = k \circ f$.

A point of surface M is said to be a regular point of f if the map f is a local diffeomorphism around the point. Otherwise, it is said to be a singular point. The singular set, denoted by $\Sigma f$, consists all singular points of f. Its image $f(\Sigma f)$ is called the branch set of f.

Throughout the paper, $\mathbb {S}^2$ will denote the unit sphere. Given an immersion $f: M \rightarrow \mathbb {R}^3$ of a closed orientable surface M into $\mathbb {R}^3$, let ${{\mathcal {N}}}_f: M \rightarrow \mathbb {S}^2$ be its Gauss map. The map ${{\mathcal {N}}}_f$ is said to be stable if there exists a neighbourhood ${{\mathcal {U}}}_f$ of f in the space ${{\mathcal {I}}}(M, \mathbb {R}^3)$ of immersions of M into $\mathbb {R}^3,$ equipped with the Whitney $C^{\infty }$-topology (see [6]), such that for all $g \in {{\mathcal {U}}}_f$, the Gauss map ${{\mathcal {N}}}_g$ associated to g is ${{\mathcal {A}}}$-equivalent to ${{\mathcal {N}}}_f$. The concept of stability for a Gauss map of M depends on perturbations of the immersion rather than on those of the map itself. Actually, for the stability of ${{\mathcal {N}}}_f$ it is equivalent to study if the family of height functions associated with f : 

$$\begin{aligned} \begin{array}{lclcl} \lambda (f) &{}: &{} M \times \mathbb {S}^{2} &{} \longrightarrow &{} \mathbb {R}\\ &{} &{} (x,v) &{} \longmapsto &{} \langle f(x),v \rangle =f_v (x) \end{array} \end{aligned}$$

is structurally stable ( [1, 2, 9]). This means that the regular points of the stable Gauss map ${{\mathcal {N}}}_f$ correspond geometrically to elliptic or hyperbolic points of M,  i.e. points where the height function in the normal direction has a stable singularity of Morse or type $A_1$. The singular points of ${{\mathcal {N}}}_f$ correspond to parabolic points of M,  i.e. points where the height function in the normal direction has a non-stable singularity. In this case, we may have: a fold point of ${{\mathcal {N}}}_f$, corresponding to a $A_2$ singularity of the height function in the normal direction or a cusp point of ${\mathcal N}_f$, when the height function in the normal direction has a singularity of type $A_3$. We assume that ${{\mathcal {N}}}_f$ is stable, in the following part.

As it was seen above, the singular set $\Sigma {{\mathcal {N}}}_f$ is the parabolic set of M associated to the immersion f. By Whitney’s theorem (see [4, 6]), the singular set of any stable smooth map between two closed orientable surfaces consists of curves of fold points, possibly containing isolated cusp points. Moreover, the image of $\Sigma {{\mathcal {N}}}_f,$ i.e. the branch set of ${{\mathcal {N}}}_f,$ consists of a collection of closed curves immersed in $\mathbb {S}^2$ with possible isolated cusps whose self-intersections (double points) correspond to parabolic points with parallel normals of the same orientation. The regular set $M-\Sigma {{\mathcal {N}}}_f,$ immersed into the surface $\mathbb {S}^2$ by the map ${{\mathcal {N}}}_f,$ consists of a finite number of connected regions.

If ${{\mathcal {N}}}_f$ is a stable Gauss map, then the number of connected components of the singular set and the topological types of the connected elliptic and hyperbolic regions are invariants, in the sense that there is a diffeomorphism that maps its singular set onto the singular set of ${{\mathcal {N}}}_g,$ when $g \in {\mathcal U}_f$ and similarly for the branch set and their corresponding regular sets, which means that topological information of a surface M is coded by these elements, so the pair $(M, \Sigma {\mathcal N}_f)$ can be reconstructed (up to diffeomorphism).

Throughout the work, we will assume that M is connected and we denote by $M^+$ the union of all elliptic regions (formed by the elliptic points of M) i.e. with positive Gaussian curvature and $M^-$ the union of hyperbolic regions (formed by the hyperbolic points of M) i.e. with negative Gaussian curvature, including their boundaries. Let us denote by $V^+$ ($V^-$) the number of connected components of $M^+$ ($M^-$) and put $V= V^++V^-.$ Denote by g the genus of M,  and $g^+$ (or $g^-$) the total genus of $M^+$ (respectively $M^-$). Clearly, $M^+$ and $M^-$ meet in $\Sigma {{\mathcal {N}}}_f.$ Finally, the number of connected components of $\Sigma {{\mathcal {N}}}_f$ is E. The following result about stable maps was shown:

Theorem 1

[7] If ${{\mathcal {N}}}_f: M \rightarrow \mathbb {S}^2$ is a stable Gauss mapping of a closed orientable surface M in $\mathbb {R}^3,$ then we have $g=1-V+E+g^++g^-.$

Fig. 1

Stable Gauss maps of the torus

Figure 1 illustrates three stable Gauss maps of the torus. The branch set of the stable Gauss map ${{\mathcal {N}}}_b$ consists of two curves each of them with 4 cusp points. The branch set of each of the stable Gauss maps ${{\mathcal {N}}}_a$ and ${{\mathcal {N}}}_c$ consists of one curve with 6 cusp points (see [2]).

3 Fold Gauss maps

In this section, we will characterize the stable Gauss maps without cusps points of closed orientable surfaces and recall a result from [8].

Definition 2

A fold Gauss map is a stable Gauss map with no cusp points.

Corollary 3

[8] If ${{\mathcal {N}}}_f: M \rightarrow \mathbb {S}^2$ is a fold Gauss map of a closed orientable surface M, then $E= 2\,(V^- - g^-).$

Lemma 4

If ${{\mathcal {N}}}_f: M \rightarrow \mathbb {S}^2$ is a fold Gauss map of a closed orientable surface M,  then all hyperbolic regions of M,  with genus zero, have at least two boundary components formed by parabolic points.

Proof

Suppose that a closed orientable immersed surface M has a hyperbolic region H with genus zero that has only one boundary component $\gamma$ adjacent to an elliptic region, then H is homeomorphic to a disc. Since M has a fold Gauss map, the zero curvature direction of M is nowhere tangent to $\gamma$, because the zero curvature direction is tangent to the parabolic curve only at cusps of the Gauss map. At every point of H, there are two lines of asymptotic directions along $\gamma$, which match with the zero curvature direction. Consider one of the two families of asymptotic lines. This is a line field on a disc that is nowhere tangent to the boundary of the disc, and the integral of this line field (a family of asymptotic curves) is a non-singular foliation of the disc that is transverse to the boundary. This is impossible because in this case the foliation has a finite number of non-degenerated singularities (proposition p. 121 on [5]), then all hyperbolic regions of M,  with genus zero, have at least two boundary components. $\square$

Theorem 5

If ${{\mathcal {N}}}_f: M \rightarrow \mathbb {S}^2$ is a fold Gauss map of a closed orientable surface M, then M satisfies that all its hyperbolic regions are homeomorphic to a cylinder bounded by circles.

Proof

Let k be the number of hyperbolic components of genus 0 of M, so $l= V^{-}-k$ is the number of hyperbolic components of positive genus. By Lemma 4, $E\ge 2k+l,$ with equality which implies each hyperbolic region of genus 0 has exactly two boundary components. By Corollary 3, $E=2(V^{-}-g^{-})\le 2V^{-}-2(V^{-}-k)= 2k$. Thus $l = 0$ and each hyperbolic region of M,  is homeomorphic to a cylinder bounded by circles. $\square$

Two immersions $f_1$ and $f_2$ of M can be connected by a path, in the immersion space $\mathcal {I}(M,\mathbb {R}^3)$, which only passes through transitions of codimension one. Some of these transitions change the number of cusps in the corresponding Gauss map and also change the topology of the singular set and, consequently, of the regular set of the immersion. From a global point of view, these transitions change the number of components of parabolic curves E and can change the number of regular component V or genus from these regions changing $g^+$ and $g^-$, but preserving the value of g. In the next section, we will see in more detail how to obtain new stable Gauss map by applying convenient codimension one transitions or surgeries to the immersions, in this sense.

4 Lips and Beaks transition and surgeries

Let $\mathcal {G}^\infty (M,\mathbb {S}^2)$ be the set of smooth Gauss maps and let $\Delta \subset \mathcal {G}^\infty (M,\mathbb {S}^2)$ be the subset formed by the non-stable Gauss maps, called the discriminant set of $\mathcal {G}^\infty (M,\mathbb {S}^2)$ which contains, for example, maps at the moment of birth or death cusps points.

We consider a generic isotopy H between two stable Gauss maps ${{{\mathcal {N}}}_f}_1,$ ${{{\mathcal {N}}}_f}_2 \in \mathcal {G}^\infty (M,\mathbb {S}^2){\setminus } \Delta :$

$$\begin{aligned} \begin{array}{lclcl} H&{}:&{} M\times [0,1] &{} \longrightarrow &{} \mathbb {S}^2\\ &{} &{} (x,t) &{} \longmapsto &{} H(x,t)=H_t(x) \end{array} \end{aligned}$$

where $H_0= {{{\mathcal {N}}}_f}_1$ and $H_1= {{{\mathcal {N}}}_f}_2.$ A codimension one transition corresponds to the intersection of a generic path $H_t$ with the set $\Delta$. In other words, the codimension one transition is the moment that the path between ${{{\mathcal {N}}}_f}_1$ and ${{{\mathcal {N}}}_f}_2$ meets transversely any strata of $\Delta$.

The codimension one transition modifies the singular set of stable Gauss maps and their effects on the stable Gauss maps that have been described in [3] by using the generic transitions in one-parameter families of height functions (defined by generic one-parameter families of immersions). According to this study, the local Morse transitions of the parabolic curve at a non-versal $A_3$ point of the height function corresponds to lips and beaks transitions.

4.1 Lips and beaks transitions

Assume that $f: M \rightarrow \mathbb {R}^3$ is an immersion of a closed oriented surface such that the corresponding Gauss map ${\mathcal N}_f$ is stable.

Definition 6

Given a cusp point x of the stable Gauss map ${{\mathcal {N}}}_f$ the singular set $\Sigma {{\mathcal {N}}}_f$ separates a small enough neighbourhood $U_x$ of $x \in M,$ into a hyperbolic region $U_x^-$ and an elliptic region $U_x^+$. Then, when any regular value $y \in \mathbb {S}^2$ of ${{\mathcal {N}}}_f\mid _{U_x}$ consists of two pre-images in $U_x^+$ and one in $U_x^-,$ or only one in $U_x^+,$ the cusp point x is called positive, in the reverse situation, when the regular value of ${{\mathcal {N}}}_f\mid _{U_x}$ consists of two pre-images in $U_x^-$ and one in $U_x^+,$ or only one in $U_x^-,$ the cusp point x is called negative.

The lips transition (see Fig. 2), that we denote by $L^\gamma ,$ corresponds to a Morse transition of maximum or minimum type in the parabolic curve, where $\gamma$ corresponds to the sign (“+” if it is positive and “-” if it is negative) of the pair of cusps that are born with the lips transition. It can be done in a region X of positive (or negative, respectively) curvature of M giving rise to a new region Z with negative (positive, respectively) curvature. Their common boundary is a connected component of the parabolic set whose image through the Gauss map is a closed curve with two cusp points in $\mathbb {S}^2$.

Fig. 2

Lips transitions

The beaks transitions correspond to a Morse transition of saddle type in the parabolic set. Such a transition occurs when we approach two arcs of the parabolic set until they join at a common point called beaks and break again, giving rise to a new pair of arcs. As a result, a couple of cusp points are introduced in the branch set. This process, from the global point of view, is to increase the cusp points. If the transition removes the cusps, we will write it with a negative sign in front.

Fig. 3

Beaks transitions: the numbers indicate components of the parabolic set

The beaks transitions (locally) can be classified into eight different cases, as illustrated in Fig. 3:

• $B_v^{+,\pm }$ increases the number $V^\pm$ of regular (elliptic or hyperbolic) connected components and the number E of curves in the parabolic set of M.

• $B_v^{-,\pm }$ decreases the number $V^\pm$ and the number E of M.

• $B_w^{+,\pm }$ increases the genus $g^\pm$ of $M^\pm$ and decreases the number E of M.

• $B_w^{-,\pm }$ decreases the genus $g^\pm$ of $M^\pm$ and increases the number E of M.

We observe that applying the beaks transitions: $B_v^{+,-}$, $B_v^{-,+}$, $B_w^{+,+}$, $B_w^{-,-}$, the sign of the pair of cusps that are born is positive. However, with the beaks transitions $B_v^{+,+}$, $B_v^{-,-}$, $B_w^{+,-}$, $B_w^{-,+}$, the sign of the cusps is negative.

Figure 1 illustrates examples of two different beaks transitions on the torus. Applying transition $B_w^{-,+}$ on the surface in Fig. 1a, which decreases the genus of the elliptic region and introduces a pair of negative cusps, we obtain Fig. 1b, and applying transition $-B_w^{-,-}$ on the surface in Fig. 1b, which eliminates two positive cusps and increases the genus of the hyperbolic region, we obtain Fig. 1c.

Fig. 4

Effects of beaks and lips transitions on surfaces and its stable Gauss maps

Example 7

Figure 4 shows examples on the sphere of two lips transitions $L^+$ and two beaks transitions: $-B^{+,+}_v,$ which eliminates two positive cusps and joins the two singular curves, and $-B^{+,-}_v$ that removes the last two positive cusps and decomposes the singular curve.

4.2 Surgeries

The operation of joining two elliptic regions of M with an intermediary hyperbolic region is called surgery on M. This process is done by removing two discs, one in each elliptic region and attaching a hyperbolic tube to their boundaries and it can clearly be done in a smooth way.

Fig. 5

Effects of surgeries $\mathcal {S}_e^0$ and $\mathcal {S}_e^1$ on surfaces and its stable Gauss maps

We denote by $\mathcal {S}_e^1$ the surgery that connects two elliptic regions of the same connected surface and by $\mathcal {S}_e^0$ the surgery that connects two elliptic regions of disjoint surfaces M and N,  called connected sum of the immersions.

Figure 5 shows the effects of the surgeries $\mathcal {S}_e^0$ and $\mathcal {S}_e^1$ on surfaces. We observe that this keeps stable the corresponding Gauss map by increasing curves in the branch set but without adding cusp points to them. The following examples illustrate some surfaces of genus $g \le 4$ associated to stable Gauss maps.

Fig. 6

Examples of surfaces with stable Gauss maps obtained by surgeries, beaks and lips transitions

Example 8

In Fig. 6 we can see examples of surfaces associated with fold Gauss maps obtained by using surgeries and some beaks and lips transitions on the sphere, considering the identity Gauss map associated to the standard embedding of $\mathbb {S}^2$ in $\mathbb {R}^3.$

• (a) Surface with two parabolic curves that separate a hyperbolic region from two elliptic regions whose fold Gauss maps is obtained by the identity Gauss map applying beaks and lips transitions on the sphere (see Fig. 4) or from $\mathcal {S}_e^0$ between two ellipsoids and their Gauss maps associated.

• (b) Surface with four parabolic curves that separate two hyperbolic region from two elliptic regions whose fold Gauss maps which can be obtained by surgeries $\mathcal {S}_e^1$ on (a).

• (c) Surface with stable Gauss map obtained by the transition $B_v^{-,+}$ followed by $B_w^{+,+}$ on (b), adding two pairs of positive cusps and joining the two elliptic regions.

• (d) Surface with Stable Gauss map obtained by using the transition $-B_v^{+,-}$, eliminating two positive cusps and joining the two hyperbolic regions on the surface (c).

• (e) Surface with fold Gauss map obtained by $-B_v^{-,+}$ on (d), which removes two positive cusps and divide the elliptic region into two, one of genus 1 and another homeomorphic to the disc.

• (f) Surface with fold Gauss map obtained by a surgery $\mathcal {S}_e^1$ on (e).

• (g) Surface with stable map obtained by the transition $B_v^{-,-}$ on (f), making two negative cusps.

• (h) Surface with fold Gauss map obtained by $-L^-$ on (g), removing the negative cusps.

• (i) Surface with fold Gauss map obtained by a surgery $\mathcal {S}_e^0$ between two copies of the surface (e).

• (j) Surface with fold Gauss map obtained by using transition $B_v^{-,+}$ followed by transition $-L^+$ on (i).

• (k) Surface with fold Gauss map obtained by surgery $\mathcal {S}_e^0$ between two copies of the Gauss map (h) followed by transitions $B_v^{-,+}$ and $-L^+,$ hence removing the hyperbolic region produced by $\mathcal {S}_e^0$.

• (l) (and (m)) Surfaces with fold Gauss map obtained by a surgery $\mathcal {S}_e^0$ between two copies of the surface (e) and by transitions $B_v^{-,-}$ and $-L^-,$ hence removing the elliptic region produced by $\mathcal {S}_e^0$.

Fig. 7

Examples of surfaces with stable Gauss maps obtained by $\mathcal {S}_{e}^1,$ beaks and lips transitions

Example 9

Figure 7 illustrates examples of surfaces obtained by surgeries $\mathcal {S}_{e}^1$ and some beaks and lips transitions.

• (a) Surface (j) in Fig. 6.

• (b) Surface with stable Gauss map obtained by applying $2(B_v^{-,+})$ on (a), creating four positive cusps and joining three elliptic regions.

• (c) Surface with fold Gauss map obtained removing the four positive cusps and joining the two elliptic regions, by transitions $-B_v^{-,+}$ followed by $-B_v^{+,-}$ on (b), which separates the elliptic region into two, one with genus 2 with a hole and another homeomorphic to a disc. Observe that the hyperbolic region is homeomorphic to a cylinder.

• (d) Surface with fold Gauss map obtained by surgery $\mathcal {S}_e^1$ on (c) adding one hyperbolic region.

• (e) Surface with stable Gauss map obtained by transition $B_v^{-,+}$ on (d), which creates two positive cusps by joining two elliptic regions and leaving the two hyperbolic regions, one homeomorphic to the cylinder and the other homeomorphic to the disc.

• (f) Surface with stable Gauss map obtained by $B_w^{+,+}$ on (e), creating two new positive cusps, by joining two parabolic curves which separate the elliptic region of genus three from two hyperbolic regions homeomorphic to the disc.

• (g) Surface with stable Gauss map obtained by the transition $-B_v^{+,-}$ on (f), which eliminates a pair of positive cusps by joining two hyperbolic regions.

• (h) Surface with fold Gauss map obtained by $-B_v^{-,+}$ on (g), removing two positive cusps. Thus the parabolic curve is divided into two components that separate one hyperbolic region, homeomorphic to the cylinder, and two elliptic regions, one with genus three and the other homeomorphic to the disc.

• (l) Surface with fold Gauss map obtained by the sequence from (h) to (l), analogous to the sequence from (c) to (h).

5 Main results

We consider a connected closed and orientable surface M which contains a subset $\Gamma$ consisting of some finite number of closed simple regular curves (embedded circles) on it. Suppose that $M-\Gamma$ is the union of regions of M,  denoted by $M_E$ and $M_H,$ and at every point of $\Gamma$ we find $M_E$ on one side and $M_H$ on the other.

Proposition 10

Let M be a connected closed orientable surface, whose subset $\Gamma$ consists of two embedded circles that separate two regions $M_E$ from one region $M_H$ homeomorphic to a cylinder. Then there is an immersion of M into $\mathbb {R}^3$ whose Gauss map is a fold Gauss map.

Proof

We begin by analysing the simplest cases.

• If the surface M has one region $M_H$ homeomorphic to a cylinder and the two regions $M_E$ are homeomorphic to the disc, then considering the identity Gauss map to the standard embedding of $\mathbb {S}^2$ in $\mathbb {R}^3$ and by using beaks and lips transitions as in Fig. 4, we obtain an immersion $f: M \rightarrow \mathbb {R}^3.$ This immersion has $f(M_E)$ as the elliptic set, $f(M_H)$ as the hyperbolic set and $f(\Gamma )$ as the parabolic set of f(M) with its Gauss map ${{\mathcal {N}}}_f$ being a fold Gauss map. Another possibility is applying $\mathcal {S}_e^0$ surgery between two spheres embedded in $\mathbb {R}^3$ to obtain the immersion with an associated fold Gauss map.

• If the surface M has one region $M_H$ homeomorphic to a cylinder and one of the regions $M_E$ has genus 1 and the other is homeomorphic to a disc, we can see in Fig. 6 the sequence (a) to (e) to obtain an immersion with this pattern of hyperbolic and elliptic points i. e. $f(M_H)$ is the hyperbolic set of f(M) and $f(M_E)$ is the elliptic set of f(M), with an associated fold Gauss map.

• If the surface M has one region $M_H$ homeomorphic to a cylinder and two regions $M_E,$ one of them with genus 2 and the other homeomorphic to a disc, the sequence (a) to (c) of Fig. 7 shows the transitions applied to obtain an immersion with an associated fold Gauss map. We obtain the surface shown in Fig. 7a by using $\mathcal {S}_e^0$ surgery between two copies of the surface in Fig. 6e (Fig. 6i) and then applying transitions $B_v^{-,+}$ and $-L^{+}$ to obtain Fig. 6j.

The immersions of these surfaces and their corresponding fold Gauss maps will serve as the base case for mathematical induction. If the surface M has one region $M_H$ homeomorphic to a cylinder and each region $M_E$ has genus 1, we can use Fig. 6m and the transition $B_v^{-,-}$ followed by $-L^{-}$ to obtain an immersion with a fold Gauss map. In the same way, if a surface M has one region $M_H$ homeomorphic to a cylinder and two regions $M_E,$ one with genus 1 and the other with genus 2, we can apply $\mathcal {S}_e^0$ surgery between the surfaces in Figs. 6e and 7c, followed by the transition $B_v^{-,-}$ and $-L^{-}$ to obtain an immersion with a fold Gauss map.

If we assume that any surface $M_i$ with two curves in $\Gamma$ and only one region ${M_i}_H$ homeomorphic to a cylinder and two regions ${M_i}_E,$ one of genus $g^+=i$ and the other homeomorphic to a disc, there exists an immersion of $f_i: M_i \rightarrow \mathbb {R}^3$ whose Gauss map ${{\mathcal {N}}}_{f_{i}}$ is a fold Gauss map and which has this pattern of hyperbolic and elliptic points, then we can prove the case for $g^+=i+1$ by considering the sequence from (c) to (h) in Fig. 7 on $M_i$ to obtain an immersion of $M_{i+1}$ with a fold Gauss map ${{\mathcal {N}}}_{f_{i+1}}.$

In the general case of surfaces M with only one region $M_H$ homeomorphic to a cylinder and two regions $M_E$ of genus ${g_1}^+=k>1$ and ${g_2}^+=l>2,$ respectively, we consider two surfaces $M_k$ and $M_l$ as in the previous case: the surface $M_k$ has only one region ${M_k}_H$ homeomorphic to a cylinder and two regions ${M_k}_E,$ one homeomorphic to a disc and the other of genus ${g_1}^+=k,$ similarly, the surface $M_l$ has only one region ${M_l}_H$ homeomorphic to a cylinder and two regions ${M_l}_{E},$ one homeomorphic to a disc and the other of genus ${g_2}^+=l.$ Then, considering the surgery $\mathcal {S}_e^0$ connecting the two regions homeomorphic to a disc of ${M_k}_{E}$ and ${M_l}_E$ and applying the transitions $B_v^{-,-}$ and $-L^{-},$ we obtain the corresponding immersion of M with an associated fold Gauss map ${\mathcal N}_{f}.$ $\square$

Proposition 11

Let M be a connected closed orientable surface of genus $g=2k,$ $k \ge 1,$ whose subset $\Gamma$ consists of 2k embedded circles that separate k regions $M_H$ each of which is homeomorphic to a cylinder from one region $M_E$ of genus k. Then there exists an immersion of M into $\mathbb {R}^3$ whose Gauss map is a fold Gauss map.

Proof

If the surface $M_1$ of genus $g=2$ has 2 curves in $\Gamma$ and only one region ${M_1}_H$ homeomorphic to a cylinder and one region ${M_1}_E$ of genus 1,  then applying one surgery $\mathcal {S}_e^0$ between two ellipsoids, and following the sequence (a) to (h) in Fig. 6, we obtain an immersion $f_1:M_1 \rightarrow \mathbb {R}^3$ with this pattern of hyperbolic and elliptic points, whose associated Gauss map is a fold Gauss map.

If we assume that any surface $M_i$ of genus $g=2i$ and $2\,i$ curves in $\Gamma$ with i region ${M_i}_H$ each of which is homeomorphic to a cylinder and only one region ${M_i}_E$ of genus i, there exists an immersion of $f_i: M_i \rightarrow \mathbb {R}^3$ with an associated fold Gauss map ${{\mathcal {N}}}_{f_i},$ then we can prove the case for $g=2(i+1)$ by considering the surgery $\mathcal {S}_e^0$ connecting the region ${M_i}_E$ of surface $M_{i}$ with the region the region ${M_1}_E$ of surface $M_1.$ By applying the transitions $B_v^{-,+}$ and $-L^{+}$ on this surface, we obtain the corresponding immersion of $M_{i+1}$ with fold Gauss map ${\mathcal N}_{f_{i+1}}$ (see Fig. 6k for the case $i=2,$ a surface of genus $g=4$). $\square$

Fig. 8

Effects of beaks transitions on surfaces of genus 1 and its stable Gauss maps

Example 12

The fold Gauss map ${{\mathcal {N}}}_e$ in Fig. 8e, can be obtained by transitions lips and beaks as follows:

• a) We consider a surface of genus 1 and four parabolic curves, which are the boundaries of two hyperbolic regions, as b) in Fig. 6, whose Gauss map is a fold Gauss map.

• b) By applying a beaks transition $B^{-,-}_v$ which joins two singular curves and two hyperbolic regions, in this case, we add two negative cusp points.

• c) Again applying a beaks transition $B^{+,-}_w$, we add two negative cusp points, join two singular curves, and introduce genus one to the hyperbolic region. Therefore, we obtain a torus with one hyperbolic region and two elliptical regions homeomorphic to the disc.

• d) First, approaches two cusp points, one on each parabolic curve, “throttling” one side of the hyperbolic region, and through a $-B^{+,+}_v$ transition, it connects the two parabolic curves by removing the pair of negative cusps, obtaining a hyperbolic region homeomorphic to the torus minus one disc.

• e) The other pair of negative cusps can be removed by a transition $-B^{+,-}_w$, “throttling” the other side of the hyperbolic region, transforming into a hyperbolic region and an elliptical region, both homeomorphic to the cylinder.

Proposition 13

Let M be a connected closed orientable surface of genus $g=k\ge 1,$ whose subset $\Gamma$ consists of 2k embedded circles that separate k regions $M_H$ each of which is homeomorphic to a cylinder from one region $M_E$ homeomorphic to a sphere minus 2k discs. Then there exists an immersion of M into $\mathbb {R}^3$ whose Gauss map is a fold Gauss map.

Proof

If the surface $M_1$ has $g=1$ and two region ${M_1}_H,$ ${M_1}_E$ both of them homeomorphic to a cylinder, then applying one surgery $\mathcal {S}_e^0$ between two ellipsoids, followed by one surgery $\mathcal {S}_e^1,$ we obtain the surface b) in Fig. 6. By using beaks transitions as in Fig. 8, we obtain an immersion of M associated to fold Gauss map ${{\mathcal {N}}}_e: M \rightarrow \mathbb {S}^2.$

If we assume that any surface $M_i$ with $g=i$ whose subset $\Gamma$ consists of 2i embedded circles that separate i regions ${M_i}_H$ each of which is homeomorphic to a cylinder and only one region ${M_i}_E$ homeomorphic to a sphere minus 2i discs, there exists an immersion of $f_i: M_i \rightarrow \mathbb {R}^3$ whose Gauss map ${{\mathcal {N}}}_{f_i}$ is a fold Gauss map, then we can prove the case for $g=i+1,$ by considering the surgery $\mathcal {S}_e^0$ connecting the region ${M_i}_E$ of surface $M_{i}$ with the region ${M_1}_E$ of surface $M_1.$ By the transitions $B_v^{-,+}$ and $-L^{+},$ we obtain the corresponding immersion of $M_{i+1}$ with a fold Gauss map ${{\mathcal {N}}}_{f_{i+1}}.$ $\square$

The preceding propositions lead us to the following result:

Theorem 14

Let M be a connected and closed, orientable surface whose subset $\Gamma$ consists of 2k,  $k\ge 1$ embedded circles. If M can be obtained as a connected sum of the surfaces with the properties described in the previous Propositions 10, 11 and 13, then there exists an immersion of M into $\mathbb {R}^3$ whose Gauss map is a fold Gauss map.