Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

Abstract

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, represented by the Riemann surfaces which intersect along the curve $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$ were addressed. Curve $ℒ$ does not contain geodesic lines in either of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Consider six points located on the $ℒ:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$/red links, and, alternatively, with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$/green links. Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

1. Introduction

Geodesic curves (abbreviated “geodesics”) are a fundamental concept in geometry used to generalize the idea of straight lines idea to curved surfaces and arbitrary manifolds [1,2]. Geodesics on smooth surfaces are the locally shortest paths [1,2,3,4]. Generally speaking, the geodesic curve is a curve representing the shortest path/arc between two points on a surface, or, more generally, in a Riemannian manifold [4]. Geodesic curves play an important role in physics [5]. The point body moving within the given surface will move along the geodesic line, if a zero total force is acting on the body [1]. In Einstein’s general relativity, geodesics in space-time describe the motion of point bodies under the influence of gravity alone [5]. In a gravitational field, the particle moves so that its world point moves along an extremal or, as it is called, a geodesic line, in the four-space [5].

We apply the Ramsey approach to the sets of points belonging to the Riemannian manifold [6,7,8,9,10]. The Ramsey theory is a sub-field of the mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large. The classical problem in Ramsey theory is the so-called “party problem”, which asks what is the minimum number of guests, denoted $R\left(m,n\right)$, that must be invited so that at least m will know each other (i.e., in the terms of the graphs theory, there exists a clique of order m) or at least n will not know each other (i.e., there exists an independent set of order n) [6,7,8,9,10]. The number $R\left(m,n\right)$ is called a Ramsey number. The graph possessing N vertices is usually denoted $K_N.$ Thus, the Ramsey theory is formulated as follows: for every two positive integers m and n, there exists a positive integer N, such that for every red-green coloring of $K_N$, there is a complete subgraph $K_m$, all of whose edges are colored red (resulting in a red $K_m$), or a complete subgraph $K_n$, all of whose edges are colored green (resulting in a green $K_n$). A classical result in Ramsey theory states that if some mathematical object/set is partitioned into finitely many parts, then one of the parts must contain a subset of a prescribed kind [6,7,8,9,10]. Within our approach, points belonging to the Riemannian manifold are connected with two kinds of links: one of which represents the geodesic curve connecting these points, and the other kind of links represents any other curve different from the geodesic line. Thus, a complete bi-colored graph emerges, and the application of the Ramsey approach becomes possible.

The suggested approach is extended to the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds: $\left(M_1,g_1\right) \mathrm{and} \left(M_2,g_2\right); \left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold $(M_1,g_1$, and, alternatively, with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$). When the graph contains six vertices, at least one closed geodesic curve inevitably appears in the graph. The problem of closed geodesics has a long history. Poincare demonstrated that on a convex surface M, there exists a simple closed curve of minimal length among the set of simple closed curves which divide M into two parts, each having total curvature 2π [11]. The existence of the closed geodesics on compact manifolds was first proved by Lyusternik and Fet [12]. Closed geodesic curves play an important role in general relativity [13].

The paper is organized as follows: first, in Section 2.1, the mathematical procedure explaining the use of the geodesic and non-geodesic lines as the links of the Ramsey bi-colored, complete graph is introduced. The suggested procedure enables the formation of the Ramsey graphs emerging from the intersection of two Riemannian manifolds, as discussed in Section 2.2. These graphs inevitably contain at least one closed geodesic line. Section 2.3 addresses bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds represented by the surfaces possessing different Euler characteristics. Section 4 puts the reported results in the context of the general problem of the closed geodesic lines.

2. Results

2.1. Geodesic and Non-Geodesic Lines Defined on a Riemannian Manifold Form a Complete Graph

Consider a smooth compact Riemannian manifold $\left(M,g\right)$, where g is the metric tensor [14]. Assume that any two points of the manifold may be connected with a geodesic curve [12]; spaces in which geodesics exist and are unique have been addressed [15]. On a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ: [a,b] → M is defined by Equation (1):

$$L\left(\gamma \right)={{\displaystyle \int }}_a^b\sqrt{g_{\gamma \left(t\right)}\left(\dot{\gamma }\left(t\right), \dot{\gamma }\left(t\right)\right)}dt$$

(1)

In Riemannian geometry, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Pestov and Uhlmann solved the inverse problem and demonstrated that through knowing the lengths of the geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with a boundary, we can uniquely determine the Riemannian metric [16].

Now, we apply the Ramsey approach to the sets of points belonging to the compact Riemannian manifold. Consider six elements belonging to manifold M: $\left\{1,\dots 6\right\}\subset M$, as depicted in Figure 1.

We connect these points with two kinds of distinguishable links; namely, the points are connected with red links $L_r\subset M$, when the line, represented by the link, is a geodesic line, and we connect the points with the green links $L_g\subset M$ when the line represented by the link is any curve different from the geodesic line. We assume that any points of the compact Riemann manifold may be connected by the geodesic or non-geodesic line, as depicted in Figure 2.

We connect the points of $\left\{1,\dots 6\right\}\subset M$ in pairs with a single colored link. Thus, the complete bi-colored graph $K_6$, depicted in Figure 1, emerges. The Ramsey number $R\left(3,3\right)=6;$thus, the graph depicted in Figure 1 inevitably contains at least one mono-colored triangle. Indeed, the triangle “135” is built only from the red links; in other words, the triangle “135” represents a closed loop built of the geodesic lines. Regrettably, the Ramsey theory tells us nothing about the specific kind of mono-colored triangle that is necessarily present in the complete bi-colored graph; this may be a mono-colored triangle built of the green or red links, i.e., non-geodesic or geodesic lines.

2.2. Bi-Colored, Complete Ramsey Graph Emerging from Intersection of Two Riemannian Manifolds

Consider two compact Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Assume that the manifolds are represented by the convex Riemann surfaces, which intersect along the following curve$:$ $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$; this is depicted in Figure 3. We also assume that curve$ℒ$ does not contain geodesic lines in both of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Imagine an intersection of the sphere with the plane, which does not pass through the center of the sphere. The emerging curve is called the “small circle”, and it does not connect the points located on it with the geodesic lines. Consider six points located on the following curve$:$ $\left\{1,\dots 6\right\}\subset ℒ$; this is shown in Figure 3.

The points $\left\{1,\dots 6\right\}\subset ℒ$ may be connected with the two distinguishable kinds of the geodesic lines; namely, they may be connected with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$, $L_r\subset M_1$, shown with the red line segments in Figure 3A, and, alternatively, they may be connected with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$, $L_g\subset M_2,$ shown schematically with the green curves in Figure 3A. The suggested procedure maps the geometrical construction shown in Figure 3A into the bi-colored, complete Ramsey graph $K_6$, depicted in Figure 3B.

According to the Ramsey theorem, this graph inevitably contains at least one mono-colored triangle. Indeed, the triangles “126”, “235”, “345” and “256” are the mono-colored red ones. Thus, they represent closed geodesic lines (loops) belonging to the Riemannian manifold $\left(M_1,g_1\right)$. Thus, we come to the following theorem:

Theorem 1. 

Consider two compact Riemannian manifolds:  $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. The manifolds are represented by the convex Riemann surfaces which intersect along the curve $ℒ:$$\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ;$ $ℒ$ does not contain geodesic lines in both of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$ . Consider six points located on the curve$:$ $\left\{1,\dots 6\right\}\subset ℒ$ . The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines; they are connected with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$; $(L_r\subset M_1)$. And, alternatively, they are connected with the geodesic lines belonging to the manifold $\left(M_2,g_2\right);\left(L_g\subset M_2\right).$ Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the graph, connected with two kinds of the links. Geodesic lines $(L_r\subset M_1)$ belonging to the Riemannian manifold $\left(M_1,g_1\right)$ are represented by the red links of the graph. Geodesic lines $(L_g\subset M_2$ ) belonging to the Riemannian manifold $\left(M_2,g_2\right)$ correspond to the green links of the graph. The emerging complete, the bi-colored graph contains at least one closed geodesic line/loop.

Proof. 

Let there be given two Riemannian manifolds: $\left(M_1,g_1\right)$ and$\left(M_2,g_2\right)$. $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ;$ $ℒ$ does not contain geodesic lines in $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, and points $\left\{1,\dots 6\right\}\subset ℒ$. Each point from $\left\{1,\dots 6\right\}\subset ℒ$ is connected with another point with the geodesic lines $L_{r }$and $L_g:$ ($L_r\subset M_1, g_1)$ and $\left(L_g\subset M_2, g_2\right).$Thus, a complete bi-colored graph $K_6$ is formed. $R\left(3,3\right)=6.$ Thus, the graph $K_6$ contains at least one mono-colored triangle $K_3$. □

The theorem predicts the existence of at least one closed geodesic line in the aforementioned construction.

2.3. Bi-Colored, Complete Ramsey Graph Emerging from Intersection of Two Riemannian Manifolds Represented by the Surfaces Possessing Different Euler Characteristics

Now we consider the particular case of the situation addressed in Section 2.2. Consider two compact Riemannian manifolds represented by the surfaces possessing different Euler characteristics, namely $\left(M_1,g_1,{\chi }_1\right)$ and $(M_2,g_2,{\chi }_2)$. Assume that the manifolds are represented by the convex Riemann surfaces which intersect along the curve$ℒ:$ $\left(M_1,g_1,{\chi }_1\right){\displaystyle \cap }\left(M_2,g_2,{\chi }_2\right)=ℒ$, as depicted in Figure 4.

Let us illustrate this suggestion with the following example: surface $\left(M_1,g_1,{\chi }_1\right)$ is a torus, i.e., ${\chi }_1=0$; the manifold $(M_2,g_2,{\chi }_2)$is represented by a surface, which is topologically equivalent to the closed disk ${\chi }_2=1$ (see Figure 4).

The points $\left\{1,\dots 6\right\}\subset ℒ$ may be connected with the two distinguishable kinds of the geodesic lines; namely, they may be connected with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1, {\chi }_1=0\right)$/torus (shown with the red curves’ segments in Figure 4A,$L_r\subset M_1;$ and, alternatively, they may be connected with the geodesic lines belonging to the manifold $\left(M_2,g_2,{\chi }_2=1\right)$/surface topologically equivalent to a closed disk (shown schematically with the green links, $L_g\subset M_2,$ in Figure 4A)). The suggested procedure maps the geometrical construction shown in Figure 4A into the bi-colored, complete Ramsey graph, depicted in Figure 4B. The emerging complete, the bi-colored graph, shown in Figure 4B, contains at least one closed geodesic line/loop. Indeed, triangles “246”, and “135” are mono-colored green ones. These triangles represent the closed geodesic lines/loops belonging to the manifold $\left(M_2,g_2,{\chi }_2=1\right)$. Thus, we come to the following corollary.

3. Corollary

Consider two compact Riemannian manifolds: $\left(M_1,g_1, {\chi }_1\right)$ and $\left(M_2,g_2, {\chi }_2\right)$. ${\chi }_1\ne {\chi }_2$are the Euler characteristics of the convex Riemann surfaces representing the manifolds. The Riemann surfaces which intersect along the curve $ℒ:$ $\left(M_1,g_1,{\chi }_1\right){\displaystyle \cap }\left(M_2,g_2,{\chi }_2\right)=ℒ;$ $ℒ$ does not contain geodesic lines in both of the manifolds$\left(M_1,g_1,{\chi }_1\right)$ and $\left(M_2,g_2,{\chi }_2\right)$. Consider six points located on the curve$:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines; they are connected with the geodesic lines belonging to the compact Riemannian manifold $\left(M_1,g_1,{\chi }_1\right)$; ($L_r\subset M_1)$. And, alternatively, they are connected with the geodesic lines belonging to the compact manifold $\left(M_2,g_2,{\chi }_2\right);\left(L_g\subset M_2\right).$Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the graph, connected with two kinds of the links. Geodesic lines $(L_r\subset M_1)$belonging to the Riemannian manifold $\left(M_1,g_1,{\chi }_1\right)$ are represented by the red links of the graph. Geodesic lines ($L_g\subset M_2$) belonging to the Riemannian manifold $\left(M_2,g_2,{\chi }_2\right)$ correspond to the green links of the graph. The emerging complete, the bi-colored graph contains at least one closed geodesic line/loop.

4. Discussion

We applied the Ramsey approach to the sets of points belonging to Riemannian manifolds, represented by Riemann surfaces. We introduced the mathematical procedure giving rise to the Ramsey complete bi-colored graphs. The procedure is based on the distinguishing between the geodesic and non-geodesic lines that connect the points of the manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic ones. This interconnection maps the points into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. If the graph comprises six points, the closed mono-colored loops (at least one loop) inevitably appear within the graph. This may be a loop built of geodesic and non-geodesic lines. The Ramsey Theorem does not predict what kind of loop/triangle will be present in the graph. This is a weak point of the Ramsey approach.

A similar approach is introduced for the Riemannian manifolds, represented by the Riemann surfaces, which intersect along the curve when the points located on the curve are seen as the vertices of the graph. The vertices in this case are connected with the geodesic lines, belonging to one of the Riemann surfaces. Thus, a bi-colored, complete graph emerges. This graph built of six vertices will inevitably contain at least one closed loop built of geodesic lines. Thus, at least one geodesic closed line will appear in the graph.

The problem of the closed geodesics is a celebrated mathematical problem [16,17,18,19,20,21]. Lusternik–Schnirelmann demonstrated that on a surface of the type of the 2-sphere $S^2$, there are three simple geodesies [18], and Fet, in turn, demonstrated that on every compact Riemannian manifold M, there is one simple closed geodesic line [19]. The state of the art in the field is summarized in [20,21]. The Ramsey approach enables a fresh glance at the problem of closed geodesics. The presented result is important for general relativity [13]. Some known solutions of Einstein’s field equations admit solutions in the form of closed timelike geodesics and closed null geodesics [13].

5. Conclusions

The application of the Ramsey analysis to the sets of points belonging to compact Riemannian manifolds is reported. The selected points are connected with two kinds of segments: geodesic and non-geodesic ones. The interconnected points are mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. According to the Ramsey theorem, the graph inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. And it remains unknown what kind of mono-colored triangle will be present in the graph. We also considered bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, represented by the convex Riemann surfaces which intersect along the curve $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$ were addressed. We assumed that curve $ℒ$ does not contain geodesic lines in both of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Imagine the intersection of the sphere with the plane, which does not pass through the center of the sphere. The emerging curve is called the “small circle”, and it does not connect the points located on it with the geodesic lines. Consider six points located on the $ℒ:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable/distinct kinds of the geodesic lines; namely, with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$, and alternatively with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$. Two points are connected with the unique link. Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the graph, connected with two kinds of links. Geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$ are represented by the red links. Geodesic lines belonging to the Riemannian manifold $\left(M_2,g_2\right)$correspond, in turn, to the green links. The emerging complete, bi-colored graph contains at least one closed geodesic line/loop. The extension to the Riemann surfaces of various Euler characteristics is therefore presented.