Flows with minimal number of singularities on the Boy’s surface

We study flows on the Boy’s surface. The Boy’s surface is the image of the projective plane under a certain immersion into the threedimensional Euclidean space. It has a natural stratification consisting of one 0-dimensional stratum (central point), three 1-dimensional strata (loops starting at this point), and four 2-dimensional strata (three of them are disks lying on the same plane as the 1-dimensional strata, and having the loops as boundaries). We found all 342 optimal Morse-Smale flows and all 80 optimal projective Morse-Smale flows on the Boy’s surface. Анотація. Розглядаються потоки на поверхні Боя. Поверхня Боя є зануренням проективної площини в тривимірний евклідів простір. Вона має природну стратифікацію, що складається з одного 0-вимірного страту (центральна точка), трьох одновимврних стратів (петель з початком в цій точці) і чотирьох двовимірних стратів (три з них є дисками, що лежать в одній площині з одновимірними стратами та мають ці петлі як свої межі). Спочатку досліджуються потоки без замкнених траекторій та з однією особливою точкою, що є 0-стратом. Показано, що у такого потоку є принамні одна сепаратриса. Доведено, що існує 18 різних структур потоків з однією сепаратрисою. Далі розглядаються потоки МорсаСмейла на поверхні Боя як на стратифікованій множині без врахування її вкладення в тривимірний простір. Доведено, що на кожному 1-страті існує сингулярна точка. Описано всі можливі (342) структури потоків, у яких 4 особливі точки (0-страт і по одній на кожному 1-страті). В кінці роботи розглядаються потоки Морса-Смейла на проективній площині, які проектуються в потоки на поверхні Боя. Такі потоки з найменшим числом особливостей мають 3 витоки, 3 стоки та 5 сідел. Описані всі можливі 80 строуктур потоків з таким набором особливостей. 2010 Mathematics Subject Classification: 37c10, 37c15, 37c20 UDC 517.956.4


INTRODUCTION
The Boy's surface is a certain immersion of the real projective plane in the three-dimensional space. It was found by Werner Boy in [2](1901). The famous mathematician David Hilbert made an assignment to Boy to prove that RP 2 could not be immersed in the 3D Euclidean space. As seen, it turned out to be possible, and this way the Boy's surface was discovered.
The first explicit parameterization of the Boy's surface was achieved by Bernard Morin in [11](1978), in which it was used to describe the sphere eversion. Another parameterization was discovered by Rob Kusner and Robert Bryant in [7](1987).
In [4](2009), Sue Goodman and Marek Kossowski showed that the Boy's surface is one of the only two possible immersions of the real projective plane with a single triple point. The advantage of the Boy's surface with respect to the Roman surface, and the cross-cap, is that it has no singularities other than self-intersections, i.e. it has no pinch-points.
We consider the Boy's surface as a stratified set. Morse theory is often used to study the topological properties of such sets and manifolds. The main constructions in it are made with the help of gradient vector fields (or flow) of Morse functions. In general position, such vector fields are Morse-Smale vector fields without closed trajectories. As the fewer singularities such vector fields have, then all constructions are simpler. Therefore, it is typical to study the structures of such vector fields with a minimum number of singularities.
The initial developments of the theory go back to [1](1937), when the physicist and mathematician A. Andronov and L. Pontryagin, respectively, considered the systemẋ = v(x), where v is a C 1 -vector field on the plane. They suggested calling it rough if for any sufficiently small perturbation in the C 1 -metric, there exists a homeomorphism in the neighborhood of the identity map, which sends orbits of the original system to orbits of the perturbed system. They also established a criteria for roughness, which is a finite number of singular points and periodic orbits, all of which hyperbolic, and no saddle connections.
One of the goals was to generalize the results. In the case of generalizing them to arbitrary oriented surfaces of positive genus, the problem was the appearance of non-closed recurrent trajectories. However, in [10](1939) A. Mayer proved that such trajectories do not exist in structurally stable flows without singularities on the 2-torus.
The topological classification of structural stable flows on a bounded part of the plane, and on the 2-spheres, were accomplished only in [8](1955) by E. Leontovich-Andronova and A. Mayer.
In [14](1959), M. Peixoto generalized the concept of roughness by the introduction of the notion of structural stability. A flow f t is called structurally stable if, for any sufficiently close flow g t , there exists a homeomorphism h sending orbits of the system g t to orbits of the system f t . Hence, the requirement of a homeomorphism in the neighborhood of the identity map (as in the definition of rough system) is relaxed for structurally stable systems.
In [15](1962), M. Peixoto showed that the concepts of structural stability and roughness are equivalent for 2D flows. As a consequence of the generalization of properties of rough flows to orientable surfaces, there arises the concept of Morse-Smale systems. In such systems, the non-wandering set consists of finitely many singular points and periodic orbits, each of which is hyperbolic, and the stable and unstable manifolds intersect transversely for any distinct non-wandering points.
Morse-Smale systems received this name after the paper, written by S. Smale, in [27](1960). He presented flows with the corresponding properties on manifolds of dimension greater than 2, and showed that they satisfy inequalities similar to the Morse inequalities, regarding the critical points of a manifold. Only in [13](1979), J. Palis and S. Smale proved that Morse-Smale systems are structurally stable.
A Morse-Smale flow without closed orbit is called a Morse flow. We say that flow is optimal if it has the lowest number of fixed points among all flow of that type on the surface.
The Morse flow on the closed surface is optimal if and only if it has only one sink and one source, according to Z. Kibalko, A. Prishlyak, R. Shchurko in [6](2018). Such a flow is also called a polar Morse flow. The topological structure of polar (optimal) Morse flows on closed 2-and 3-manifolds was described in [3,5,6,9,17,18,21].
The formula for the sum of singularities indices is useful for calculating their number for flow on a stratified set [19].
In this paper we unit these two important concepts in mathematics (Boy's surface and Morse-Smale systems) that have being developed since the beginning of the XX century. One of the goals here is to describe topological structures of flows with minimum number of singular trajectories.
Structure of the paper. This paper contains results concerning flows applied to the Boy's surface with minimal number of singularities. We prove Theorem 3.1 that states that there exist 80 optimal PMS-flows in the projective plane, with 3 sources, 5 saddles, and 3 sinks.

SINGLE FIXED POINT FLOWS
Let us describe some topological properties of the Boy's surface. The Figure 1.1 is the so-called fundamental (curved) polygon of the Boy's surface [2]. In order to show that, the following two remarks are helpful: Consider the Boy's surface ( Figure 1.1 right), which is the immersion of the projective plane in three-dimensional space, and cut it by its 3 loops in order to obtain 4 disks. The loops are the 1-strata, and the disks are the 2-strata.
Since the triple point (0-stratum) is unique, when cutting the 3 loops, 6 copies of the same point are generated (one in each loop and one in each boundary connecting loops -red points in Figure 1  Now, let's consider the proof about the 0-stratum. As the vector field on the Boy's surface is assumed to be smooth, each of its vectors belong to the tangent space at a point. When such point is a 0-stratum (point of intersection of 3 surfaces), the vector at this point must be tangent to all 3 surfaces simultaneously, which is possible only when this is a zero vector. Therefore, the vector at the 0-stratum is zero for every flow, and consequently the point is singular. In order to visualize it, make a third surface intersecting the two planes in Figure 1.2, crossing the red line. This triple intersection point is the singular 0-stratum point. □ Consider flows with one fixed point being the 0-stratum. It is easy to see that the α-and ω-limit set of each trajectory is the 0-stratum. If this is not so, then, according to the Poincare-Bendixson theorem, there must exist a closed cycle and a second fixed point inside it.
The 1-stratas and the separatrices divide the surface into regions, which we will call cells. Each cell is a curved polygon, the vertices of which are fixed points. Since there are no fixed points inside the cell, each trajectory starts and ends at one of the vertices. There are 4 types of vertex angles: elliptic, hyperbolic, sources, sinks.
‚ In a neighborhood of the elliptic point, the vector field is topologically equivalent to the vector field tx 3´3 xy 2 , 3x 2 y´y 3 u, x ě 0, y ě 0; ‚ in the neighborhood of the hyperbolic point it is equivalent to to the vector field tx,´yu, x ě 0, y ě 0; ‚ near the source to tx, yu, x ě 0, y ě 0; ‚ and near the sink to t´x,´yu, x ě 0, y ě 0.
Also note that all cells are of two types: ‚ one of the corners is a source, another corner is a sink, and the rest are hyperbolic (polar cell); ‚ one of the angles is elliptic, and the rest are hyperbolic (cyclic cell).
In a polar cell, trajectories start and end at distinct points, whilst in a cyclic cell, the trajectories start and end at the same point. The boundary of the cyclic cell forms a cycle, and the polar one forms two oriented paths that start at the source and end at the sink. Of course, for MS-flows elliptic vertex angles is not allowed, and thus these flows only have polar cells.   In case a) the arcs will be oriented as follows: In case b) we have the following orientations of the arcs: In case a) we have three angles that are potential sources (p-source): 3, 6, 9 and three angles that are potential sinks (p-sink): 2, 5, 8. If a separatrix enters the potential source, then it turns into a saddle. Similarly, if a separatrix starts from the potential sink, then it also turns into a saddle.
In other cases, they remain sources and sinks, respectively. Note that the separatrix cannot start in the source and go to the sink, since in this case it will be a regular curve.
Let us consider case a) in detail. Suppose there is a separatrix connecting two potential sinks, for example, 2 and 5. Then it splits the region into two parts in one of which there will be two potential sinks 6 and 9, which means there is a another separatrix that connects them or a separatrix that separates them into different area.
Thus, in this case there are at least two separatrices. Similarly, if there is a separatrix that connect to sources, then there exist an other separatrix. Now consider the case of separatrix, that starts in a p-sink and go to a p-source. If this points are opposite (3 and 8, 2 and 6, or 5 and 9) then we obtain to polar regions. So the structure of the flow is determined by the separatrix. If the separatrix connect not opposite p-sink and p-source, then one of region contain two p-sink and two p-source and there is other separatrix in it.
Theorem is completed. So each angle in a 0-strata is hyperbolic, sink or source. The structure optimal MS-flows has the minimal number of critical elements, among all MS-flows.

Definition 2.3. A separatrix, in the context of Morse-Smale dynamical systems, is a trajectory connecting either a saddle to a sink, or a source to a saddle and belonging to a 2-strata.
Connections between saddles, or between a sink and a source, are forbidden in Morse-Smale systems. Since the definition of optimal MS-flow includes the requirement of having minimal number of singular points, then an optimal MS-flow cannot have more than 4 singular points. Therefore, a system with 5 singular points (4 in the boundary and 1 in the largest 2-stratum, for example) is not optimal. Using this fact, we will prove the theorem by contradiction.
Suppose that there exists a closed orbit on the Boy's surface. By definition of an MS-flow this must be a hyperbolic cycle. It follows from the Brouwer's fixed point theorem that there exists at least one singular point in the interior of the compact region formed by this cycle. But then this singular point belongs to a 2-stratum, which adds up to 5 singular points on the whole Boy's surface (see Figure 2.1 right). This contradicts the requirement that an optimal MS-flow cannot have more than 4 singular points. Hence, there are no optimal Morse-Smale systems with closed orbits (hyperbolic cycles) on the Boy's surface. □ Proof. We will prove it by contradiction. Suppose that the orientation in the boundary of the curved polygon in Figure 2.1 is changed such that the flow leaves point 11, go around the loop, and end at 9, i.e. the point 10 is regular. Then the flow goes from 9 to 8 and from 12 to 11, or vice-versa. In each case, the regular point 9/11 and the point 1 are identified together. This implies that if 10 is a regular point, then 1 must be regular as well. This is a contradiction for 1 is a 0-stratum, and thus it is a singular point. Therefore, in every loop there is at least one singular point (sink, source or saddle). . Since singular points lying on 1-strata have two trajectories entering them along the 1-stratum, they cannot form a red saddle with an incoming separatrix. Thus, in our case, the flow does not have red separatrices. Let us prove that such a flow has a unique sink. Suppose there are 2 sources. Then, after cutting the fundamental region along the green separatrices, due to its connectivity, a region will be formed that has 2 sinks on the boundary. Then they must be separated by a red separatrix, which is impossible. The resulting contradiction proves the existence of a unique sink. There are two different possibilities for choosing such a point -these are points 2 and 4.  We begin our consideration of such flows with flows without red separatrices (Figure 2.3-11). In this case, considering the arguments as above, there is only one sink. It can be one of points 2, 4 or 8. The remaining 3 points (12,16,18) give symmetrical flows with respect to the first 3. Hence, there are 3 topologically non-equivalent flows without red separatrices.
Consider now the case of one red separatrix. It can end at the saddle 6 or 10 (case 14 is symmetrical to 6). Let the separatrix end at 6. If at the same time it starts at 3 (3 Ñ 6, see Figure 2.3-5), then 4 is a sink, and in another area into which it divides the diagram, the remaining green points can be sinks: 2, 8, 12, 16, 18. So there are 5 different threads in this case.
If the separatrix starts at 1 (1 Ñ 6, Figure 2.3-3), it divides the area into two parts with two (2, 4) and four (8,12,16,18) points that can be sinks, thus 2ˆ4 = 8 options in total. In case 17 Ñ 6 ( If the red separatrix ends at 10, then possible axial symmetries of the diagrams must be taken into account. Thus, for example, all options with 6 Ñ 10 are symmetrical to the corresponding options with 14 Ñ 10. Therefore, we will consider only one of them. . Summing up all these numbers, we get 5 + 8 + 9 + 8 + 5 + 5 + 8 + 6 = 54 variants with one red separatrix. We now turn to the case with two red separatrices. Red saddles can be 6 and 10 or 6 and 14 (the case of 10 and 14 is symmetrical to 6 and 10). In this case, the number of variants will be equal to the product of the numbers of saddles lying in three parts into which two red sepraartris divide Points 4, 10 and 16 are either sink (for every 2-strata) or saddle (for every 2-strata), since these are inner points on the projective plane. Of course, the boundary orientation in which all the potential sources are potential sinks, and vice-verse, is also possible, but the results will be similar. The first portrait shows the case in which 4, 16 are saddles, and 10 is a sink. This is the only possibility of green separatrices, up to symmetry, for optimal PMS-flow.
When 4, 16 are sinks, and 10 saddle, there will necessarily be a saddle (yellow point) inside the largest 2-stratum. This is represented in the second and third portraits. The dashed red lines are the possible red separatrices. We have 6 cases in the first portrait and 3 in the second, i.e. 9 cases.
The remaining portraits present the situation in which 4, 10, 16 are sinks. There will be two saddles inside the largest 2-stratum. Counting all Suppose a trajectory leaves the 0-stratum. Then, since the optimal projective flow is symmetric at fixed points, the 0-stratum is a source for all angles in it. Each 1-stratum has one fixed point of the p-sink (potential sink). Let us enumerate the corners as in Figure 2.2 left. Inside each 2-disk the following two types of flow are possible: with a sink inside or with a sink at the boundary (in the p-sink). The 0-stratum corresponds to three points on the projective plane. Therefore, the flow will have at least three sources. Since there must be a sink on each 2-disk, there are also at least 3 sinks.
Further we will show the existence of flows with three sources and three sinks. Since the Euler characteristic of the projective plane is equal to 1 and is equal to the sum of the number of sources and sinks minus the number of saddles, the optimal flow has three sources, three sinks and five saddles. If all the sinks of the 2-disks are internal, then the central region must also have a sink, and therefore such a flow will not be optimal.
Let only one disk have a sink at the boundary. For definiteness, assume that its index is 10. Then a unique structure of such a flow is possible. In this case, all green separatrices end at 10.
Next, consider the case of two sinks at the boundary. Assume that their indexes are 4 and 16. For such flows there possible an axial symmetry transforming 4 into 16. These points should be separated by red separatrices. Since all red points on the border are sources, there is a saddle inside the central region. The two separatrices that enter into that region start at points that lay on opposite sides of points 4 and 16. Then one of them is 1, 3, or 17, and the other is 5, 7, 9, 11, 13 or 15. If the first is 1, then 5, 7, 9 are symmetric to 15, 13, 11, respectively. In other words, there are only three different structures of such flows. Since 3 is symmetric to 17, we will consider only the options of separatrices starting at 3. For the second separatrix, 6 options are possible, and thus we have 6 more flow structures. Now we consider the case of three sinks at the boundaries of 2-disks, that is, when points 4, 10, and 16 are sinks. In this case, there are two saddles inside the central area. Their stable manifolds (red separatrices) are separated by points 4, 10, and 16. There are 30 structures of such type. To see this, we introduce the concept of saddle weight. The stable manifold of the saddle splits the central region into two parts. The weight is equal to the number of saddles +1 of the part in which there is one sink.
In view of the symmetries, we can assume that these regions for a pair of saddles contain sinks 4 and 16 and the weight of the saddle adjacent to the first region (with sink 4) is not greater than the weight of the second saddle. Then the numbers of structures for distinct weights is given in the following

CONCLUSIONS
It is known that there is a unique optimal Morse-Smale flow on the projective plane and it has 3 fixed points: one source, one sink and one saddle. We have shown that the condition that the flow is projected into a flow on the Boy's surface increases the number of fixed points to 11, and the possible flows themselves to 80. If the Boy's surface is considered as a stratified set, then there are even more optimal flows, namely, 342. But if we do not impose Morse-Smale conditions, then there are only 18 optimal flows. A remained interesting problem consists of describing of all possible extensions of the constructed PMS-flows to optimal flows on a three-dimensional sphere, as well as describing optimal flows for immersions of other surfaces.