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A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature

We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral.


introduction
We start with the definitions of the specific curvature of a domain W and the Gaussian curvature at a point P on a convex surface which is considered as the whole boundary of a convex body in R 3 , and mention some fundamental results on a complete convex surface of bounded specific curvature from the book of A.D. Alexandrov (see [1]).
Definition 1 (see [1, pp. 365-366]). Any domain W on a convex surface has some curvature ω(W ) = W K(P ) dS, where K is the Gaussian curvature at the point P and S(W ) is the corresponding area. The ratio ω(W ) S(W ) is called the specific curvature of the domain W , and it is denoted by κ(W ). A convex surface has Gaussian curvature equal to K at a point P if the limit of the specific curvature of the domain tends to the limit K whenever this domain shrinks to the point P .
We recall the following fundamental results which have been formulated and proved by A.D. Alexandrov (see [1, pp. 365, 377: Theorems 2, 3, p. 7, 64-66: Theorem 3, p. 377: footnote 9]): (1) If the specific curvature κ(W ) of every domain W lying in some neighborhood of a point P on a convex surface does not exceed some positive number K then there exists r 0 > 0 such that it is possible to draw a shortest arc of length at least r 0 from the point P in each direction.
(2) Suppose that a specific curvature κ of a convex surface M is bounded on a neighborhood W , i.e., K 1 ≤ κ(W ) ≤ K 2 . Introduce the polar geodesic coordinates on the sphere S Ki (i = 1, 2), and construct a mapping such that to each point of the neighborhood W with coordinates r, ϕ, correspond points of the spheres S Ki for i = 1, 2, with the same coordinates. If, to a curve L on M , this mapping puts in correspondence the curves L 1 and L 2 on the spheres S K1 and S K2 , then the lengths of these curves are related by the inequalities S(L 1 ) ≥ S(L) ≥ S(L 2 ).
(3) If the specific curvature κ(W ) is ≥ K (≤ K) in a triangle △ABC, then the angles of △ABC are no less (or greater) than the corresponding angles of the triangle △ABC on the K-plane (see for the definition of K-plane in Section 3, p. 9). For K > 0, we consider the perimeter of the triangle to be lower than 2π √ K . (4) Each two points of a complete convex surface can be connected by a shortest arc.
(5) If the specific curvature κ(W ) is ≤ K on a complete convex surface, then each arc of a geodesic of length at most π √ K is a shortest arc in general but not compared with close lines.
We consider M to be a complete convex surface with bounded specific curvature K 1 < κ(W ) < K 2 . The length of each shortest arc on W is greater than π √ K2 and smaller than π √ K1 . We state the weighted Fermat-Torricelli (w. F-T) problem on M for quadrilaterals. Problem 1. Let ABCD ⊂ M be a convex quadrilateral whose perimeter is less than 2π √ K1 . Suppose that a positive number (weight) w R , corresponds to the vertex R ∈ {A, B, C, D}. Find the w. F-T point P F such that f (P F ) = w A l A + w B l B + w C l C + w D l D → min (1.1) where l R is the length of the shortest arc from the w. F-T point P F to the vertex R ∈ {A, B, C, D} (Fig. 1).
In the paper we provide the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature M (Theorem 1 and Corollary 1) by applying a method of differentiation for shortest arcs which has been introduced for the differentiation of the length of geodesics in [11], [10] which generalize the plasticity equations introduced in [9] on the two dimensional K-plane (Two dimensional sphere, hyperbolic plane and Euclidean plane) and we prove the plasticity property of convex quadrilaterals on M which was numerically verified for convex quadrilaterals on the two dimensional K-plane in [9] without giving a proof.
The main result of the paper is the plasticity principle of convex quadrilaterals on M (Theorem 2 in Section 2) which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point and their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc.
Using comparison geometry, we derive the plasticity equations for convex quadrilaterals for two cases of manifolds with metric of positive curvature which has been obtained by the gluing theorem of A.D. Alexandrov (Corollary 2 in Section 3). These results provide a generalization of the evolution of convex quadrilaterals which has been introduced in [ for which P minimizes where l R is the length of the shortest arc from P to the vertex R ∈ {A, B, C, D}. This is the 4-inverse w. F-T problem on M (Fig. 1).
For w D = 0, we derive the 3-inverse w. F-T problem which has been introduced and solved in R 2 by S. Gueron and R. Tessler in [5] and it has been solved by [8] on a surface of constant Gaussian Curvature and generalized in [10], [11] on a C 2 surface. The solution of the 3-inverse weighted Fermat-Torricelli problem gives a positive answer on this problem on a C 2 surface. It is called the 3-inverse normalized weighted Fermat-Torricelli problem for c = 1 (see [5], page 449). We show that a solution of the 4-inverse weighted Fermat-Torricelli problem ( Corollary 1) gives a negative answer to this problem because three weights depend on one variable weight ( Theorem 1).
An application of the solution of the 4-inverse w. F-T problem is the derivation of the evolution of convex quadrilaterals on M . The evolution of convex quadrilaterals is given by the invariance of the weighted Fermat-Torricelli point for given convex quadrilaterals (geometric plasticity) and the plasticity of convex quadrilaterals which are convex quadrilaterals which satisfy Theorem 1 (dynamic plasticity). We note that the geometric plasticity was known to Viviani for the case of n given points in R 2 . Theorem 1. Consider the 4-inverse w. F-T problem on a complete convex surface of bounded specific curvature M in R 3 . The plasticity of convex quadrilaterals on M is given by the following three equations:

2)
and The weight (w R ) ABCD corresponds to the vertex R that lie on the shortest arc P F R, for R ∈ {A, B, C, D} and the weight (w S ) SLN corresponds to the vertex S that lie on the shortest arc P F S regarding the triangle △SLN , for S, L, N ∈ {A, B, C, D} and S = L = N.
Proof of Theorem 1: Firstly, we assume that we choose four initial (given) values (w R ) ABCD (0) concerning the weights (w R ) ABCD for R ∈ {A, B, C, D} such that the w. F-T point P F exists and it is located at the interior of ABCD. Differentiation of the objective function with respect to a specific arc length, yields the system Without loss of generality, we choose Q ∈ {A, B, C, D}. We suppose that the point Q, the arc length parameterized curve c(t) and the point P F that lie in c(t) are given in the neighborhood W PF . We apply the following general result given in [4, Corollary 4.5.7, p. 125, Remark 4.5.8, p. 126 and Theorem 4.5.6, p. 123] that deals with a rule for differentiating the length of a continuous family of shortest arcs connecting Q to points c(t), for Q ∈ {A, B, C, D} and the existence of the right derivative d dt l Q , although the shortest arcs from Q to c(t) may not be unique on a non-negatively curved complete locally compact space X (M is a subset of X): Let c(t) : [0, T ] → M be arc length parameterized geodesic, Q a fixed point in M , Q = c(0). Then the function t → l Q (t) = |Qc(t)| has the right derivative and where ϕ ′ c,min (t) is the infimum (minimum) of angles between c(t) and the shortest arcs connecting c(0) to Q and takes a unique value ϕ ′ c (t) on M. We choose the parametrization (a similar technique has been used in [11], [10] for the parametrization of length of geodesics): We assume that the shortest arcs l B , l C , l D , can be expressed as functions of l A : From (2.6) and (1.1) the following equation is obtained: Taking into account that the point P F is the w. F-T point of ABCD we derive that the right derivative of (2.7) with respect to the variable l A and considering (2.5), we get (2.8) From (2.4) and (2.5), for Q = B, we get Let the point C, the length parameterized curve c(t) and the point P F that belongs in c(t) be given in the neighborhood W PF . Taking   Replacing (2.9) -(2.11) in (2.8), we obtain Similarly, working cyclically and differentiating (1.1) with respect to l B , and choosing the parametrization l B (t ′ ) = t ′ , by differentiating (1.1) with respect to l C and by choosing the parametrization l C (t ′′ ) = t ′′ and by differentiating (1.1) with respect to l D and choosing the parametrization l D (t ′′′ ) = t ′′′ , we derive three relations, respectively, (2.14) The equations (2.12) -(2.15) could be written in a unified form The points of M are distinguished into three categories (see [1, p. 386]): (1) conical points at which the tangent cone has complete angle less than 2π, (2) edge points at which the tangent cone is dihedral angle, (3) "smooth" points at which the tangent cone is a plane.
The point P F which is the intersection of four prescribed shortest arcs in M can be only a smooth point and it cannot be a conical point or an edge point, due to a result of A.D. Alexandrov in [1, Theorem 5, p. 135]. Therefore, the complete angle of P F is given by the following formula: (2.16) Taking into account the 4-inverse w. F-T problem, the weights (w R ) ABCD = w R for R ∈ {A, B, C, D} become variable weights which satisfy (2.12) -(2.15). Taking into consideration the trigonometric identity and the orientation of the angles with respect to P F taken counterclockwise (see Fig. 1) we solve the linear system (2.12) -(2.13) with respect to w A and w B : Similarly, solving the linear system (2.12) and (2.14) with respect to w A , w C , we obtain We write (2.19) in the following form: If we set (w D ) ABCD = 0, in (2.17), (2.18) and (2.19), we obtain a solution of the 3-inverse w. F-T problem for the triangle △ABC (see [10], for a solution of this problem on a C 2 -regular surface) and we get Similarly, if we set (w B ) ABCD = 0, in (2.17) -(2.19), we obtain a solution of the 3-inverse w. F-T problem for the triangle △ACD ⋆ , where (D ⋆ is the symmetric point of D with respect to P F , and we get which gives (2.1). We proceed by deriving (2.2). We write (2.17) in the form ).
If we set (w D ) ABCD = 0, in (2.17) -(2.19), we obtain a solution of the 3-inverse w. F-T problem for the triangle △ABC and we get Similarly, if we set (w C ) ABCD = 0, in (2.17) -(2.19), we obtain a solution of the 3-inverse w. F-T problem for the triangle △ABD), and we get which yields (2.2).
As a direct consequence of Theorem 1 we obtain We continue by proving the main result, which we call a plasticity principle of convex quadrilaterals on M : Theorem 2. Given four shortest arcs which meet at the weighted Fermat-Torricelli point P F and their endpoints form a convex quadrilateral on M and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight that corresponds to a shortest arc causes a decrease to the two weights that correspond to the two neighboring shortest arcs and an increase to the weight that corresponds to the opposite shortest arc.
Proof of Theorem 2: We take into account the three plasticity equations of Theorem 1 and Corollary 1 which are derived by applying the conditions ABCD w := (w A ) ABCD (1 + wB wA + wC wA + wD wA ) ABCD and ABCD w = ABC w = ABD w = ACD w = BCD w = const such that (2.20) -(2.21) hold. We prove that a A , a C < 0 and a B > 0. We calculate the coefficient a A : . Taking into account that P F is located at the interior of the triangle △BC ⋆ D, we have Concerning the triangle △BCD, P F is not located at the interior of P F and by the relation Therefore, because P F is located at the interior of the triangles △ABC, △ABD, which makes the weights . (2.24) Taking into account the counterclockwise orientation of angles with respect to P F (see Fig. 1), we get Similarly, taking into account the counterclockwise orientation of angles with respect to P F (see Fig. 1), we get Therefore, we have that completes the proof.
A numerical verification of the plasticity principle is given in [9, Examples 4.7, 4.10, pp. 418-419]. Remark 1. Concerning the plasticity of convex quadrilaterals on M , we have assumed that P F is located at the interior of △ABC. Then a fourth shortest arc "grows" from P F such that its length reaches l PF (D) and ABCD is a convex quadrilateral on M . Therefore, we consider that the 4-inverse w. F-T problem is derived by the 3-inverse w. F-T problem on M .
We mention two particular cases (Proposition 1, Example 1) such that P F belongs to at least one of the diagonals (shortest arc) of ABCD where the 4-inverse w. F-T problem is not derived by the 3-inverse w. F-T problem like in Theorem 2.
Replacing (2.25) to the inverse condition Solving (2.26) and (2.27) with respect to w C and w A , we derive that where x C , y C , x B , y B are constant numbers such that: and Det is the determinant of (2.26) and (2.27): We will prove that x C < 0 and x B > 0. The determinant Det of (2.26) and (2.27) is a negative because sin(∠CP F B) < 0, sin(∠BPF C) sin(∠BPF A) < 1, sin(∠BP F A) < 0, sin(∠CP F A) > 0, sin(∠BP F C) > 0. Taking into account that the numerator of (2.28) we obtain that x C < 0. Taking into account that Det< 0 and that the numerator of (2.29) and that because sin(∠BP F A) < 0 (If P ′ is the point where the diagonals meet we take P F ∈ [B, D] to be located closer to B with respect to P ′ ), we obtain that x B > 0.
The following equations where x C < 0 and x B > 0, give the plasticity principle of ABCD, because by increasing w D , the weight w C will be decreased and the weight w B will be increased and considering the relation (2.25) w A will also be decreased because − sin(∠BPF C) is a positive number. From (2.17), (2.19) we derive w A = w C and w B = w D , respectively. Replacing these two relations in the 4-inverse condition of Problem 2, we have where c is a positive real number. By the last relation, an increase in w D = w B will cause a decrease in w A = w C , otherwise the inverse condition will not hold.

Remark 2.
We would like to mention a reformulation of the 4 inverse w. F-T problem for a convex quadrilateral ABCD on R 2 , which is communicated to the author by Professor Dr. Vladimir Rovenski. Let Π = {w A + w B + w C + w D = 1} (say, c = 1) be a 3-plane in the linear space R 4 of variables (w A , w B , w C , w D ). Denote v 1 = P A/|P A|, v 2 = P B/|P B|, v 3 = P C/|P C|, v 4 = P D/|P D| -the unit vectors in R 2 . The linear system w A v 1 + w B v 2 + w C v 3 + w D v 4 = 0 (of two equations with respect to (x, y) given coordinates) determines a straight line w R (t) = w R (0) + t e in Π, where e = (e 1 , e 2 , e 3 , e 4 ) is a direction vector of the line.
Taking the inner product of w A v 1 + w B v 2 + w C v 3 + w D v 4 = 0 with respect to the vector v i for i = 1, 2, 3, 4, we obtain (2.12) -(2.15).
Taking the exterior product of w A v 1 + w B v 2 + w C v 3 + w D v 4 = 0 with respect to the vector v i for i = 1, 2, 3, 4, we obtain (2.17) -(2.19).

The plasticity of quadrilaterals on a manifold of positive curvature
We give some evolutionary structures for quadrilaterals on a manifold with positive curvature. A metric of positive curvature is intrinsic and the sum of the lower angles of every sufficiently small convex triangle cannot be less than π (see [1, p. 279]). The evolutionary structure for quadrilaterals is derived by the plasticity equations and the 4-inverse Fermat-Torricelli problem for the following two cases: (1) A manifold M ′ which is obtained by gluing two triangles that exist on a two-dimensional sphere with constant Gaussian curvature K 1 and two triangles that exist on a two-dimensional sphere of constant Gaussian curvature K 2 , for K 1 < K 2 , and the sum of angles meeting at the Fermat-Torricelli point P F of the quadrilateral equals 2π, (2) A manifold M ′′ which is obtained by gluing two triangles that exist on a complete convex surface of bounded specific curvature M (K 1 < κ < K 2 ) and two triangles that exist on a two-dimensional sphere of constant Gaussian curvature K 1 , and K 2 , respectively, and the sum of angles meeting at the Fermat-Torricelli point P F of the quadrilateral equals 2π.
These two cases form a manifold with metric of positive curvature because they are specific cases of the gluing theorem of A.D. Alexandrov (see [1,

pp. 278-279]) which states that:
If a manifold is obtained by gluing a finite number of polygons with metric of positive curvature so that, at each vertex, the sum of angles of these polygons meeting at it is no greater than 2π, then the metric on the whole manifold is also a metric of positive curvature.
A further generalization of the gluing theorem of A.D. Alexandrov was made by Yu. Reshetnyak concerning the gluing of Cartan-Alexandrov-Toponogov spaces with curvature (in the sense of A.D. Alexandrov) bounded from above by a real number K 2 (CAT(K 2 )) spaces along proper convex subsets is a CAT(K 2 ) space (see [7, pp. 188-189 and footnote 15] and [2]).
We mention the definition of a K 0 -plane and the comparison triangle of a triangle on a two-dimensional surface.
If K 0 denotes the constant Gaussian curvature of a surface M , then M is called the K 0 -plane.
If K 0 = 0, the K 0 -plane is an Euclidean plane R 2 .
A comparison triangle on a K 0 -plane denoted by (△ABC) K0 of △ABC on a two-dimensional surface is a triangle whose corresponding sides have equal lengths: pp. 185, 188]). The existence of the comparison triangle △(ABC) K0 , for K 0 > 0, is given by the condition that the perimeter of the triangle is not greater than 2π √ K0 .
The following corollary is given on M ′ and M ′′ which are particular cases of M considered in Theorem 1.

Corollary 2. Consider the 4-inverse w. F-T problem on M q for q ∈ {′, ′′.}
The following equations point out the plasticity of quadrilaterals A q B q C q D q and on M q :

2)
and where the weight (w R ) A q B q C q D q corresponds to the vertex R that lie on the shortest arc P F R, R ∈ {A q , B q , C q , D q } and the weight (w S ) SLN corresponds to the vertex S that lie on the shortest arc P F S regarding the triangle △SLN , for S, L, N ∈ {A q , B q , C q , D q } and S = L = N, for q ∈ {′, ′′}.

Proof of Corollary 2 (Case M ′ ): Gluing the comparison triangles (△A
and △BP F A, respectively, we obtain the following angular relations on M ′ : (where ǫ i , are non-negative real numbers for i = 1, 2, 3, 4) such that The angular relations hold due to a result of A.D. Alexandrov also known as the angle comparison theorem which states that: Let T be a triangle on a complete convex surface of bounded specific curvature and let triangle T 1 , T 2 be triangles on the two-dimensional spheres S K1 and S K2 with sides of the same length as T . If α, α 1 , α 2 are corresponding angles of these triangles then α 1 ≤ α ≤ α 2 (see [1,Theorem 4,p. 377, p. 54, case 1]).
We apply the following general result which was given in [4, Corollary 4.5.7, p. 125, Remark 4.5.8, p. 126 and Theorem 4.5.6, p. 123] that deals with a rule for differentiating the length of a continuous family of shortest arcs connecting Q to points c(t), for Q ∈ {A, B, C} and the existence of the right derivative where ϕ ′ c,min (t) is the infimum (minimum) of angles between c(t) and the shortest arcs connecting c(0) to Q. Taking into considerations the parameterization l Q (t) = t, for Q ∈ {A ′ , B ′ , C ′ , D ′ } and differentiating the objective function (1.1) with respect to t and following the same process in the proof of Theorem 1, we get Applying the same steps that have been used in the proof of Theorem 1, we obtain (3.1) and (3.2). This system of equations induces a comparative plasticity between the plasticity of ABCD on M and A ′ B ′ C ′ D ′ on M ′ because we may compare the plasticity equations on a convex surface M with the plasticity equations on a convex surface M ′ such that the given directions with respect to P F vary in a specific way that depend on ∠R ′ P F S ′ for R ′ , S ′ ∈ {A ′ , B ′ , C ′ , D ′ } : and every ratio For instance by taking into account that and let a A , a C be increased by δa A , δa C > 0 then a B will be decreased by δa B < 0 such that where a R is taken by Corollary 1, for R ∈ {A, B, C}.

Proof of Corollary 2 (Case M ′′ ):
Taking into consideration the gluing of the triangles △AP F D, △DP F C which exist on M with the two comparison triangles respectively, we obtain the following angular relations on M ′′ : (where ǫ i are non-negative real numbers for i = 3, 4) such that The angular relations hold due to a result of A.D. Alexandrov also known as the angle comparison theorem (see [1, Theorem 4, p. 377, p. 54, case 1]). Applying the same process that have been used in the proof of Theorem 1, we derive the desired plasticity equations (3.1) and (3.2).
Remark 3. The comparative plasticity between the surfaces M and M ′ or M ′′ given in Corollary 2 approaches some type of formulation of the plasticity of a mathematical "gibbosity" (part of a sphere on a convex surface) and it might be considered as a useful tool for future medical applications in the area of robotics.

A symmetrization of weighted quadrilaterals on a surface of Gaussian curvature bounded above by a positive number
We introduce a new symmetrization technique which transforms some classes of weighted convex quadrilaterals on a C 2 complete convex surface of bounded Gaussian curvature M to parallelograms which lie on the same tangent plane that is defined at the weighted Fermat-Torricelli point of the corresponding quadrilateral.
We give two classes of parallelograms which characterize the evolution of convex quadrilaterals which means that the weights satisfy the plasticity equations of Theorem 1 The variable Gaussian curvature K is positive and bounded below by a real positive number K 1 : K 1 < K, then the geodesic distance of length not greater than π √ K1 and is unique, and an estimate of the injectivity radius r i = inf (r i (P ) : P ∈ M ) of M is given by the inequality r i ≤ π √ K1 , see [6], [7, Theorem 3.5.2]. We consider a convex quadrilateral A • B • C • D • that belongs on a neighborhood W A ⊂ M , where W A is a subset of a geodesic circle with center A and radius r i , and the perimeter of the quadrilateral is smaller than 2 r i .
Suppose that we select w R for R ∈ {A • B • C • D • } (Fig. 2) such that P F is located at the interior domain of A • B • C • D • and some inequalities are also satisfied We consider a quadrilateral A • B • C • D • that belongs on a neighborhood W A ⊂ M , where W A is a subset of a geodesic circle with center A and radius r i , and the perimeter of the quadrilateral is smaller than 2 r i . Figs. 2 and 3), From (4.1) and (1.1) the following equation is obtained replacing Let c(t) : [0, T ] → M be a geodesic parameterized by arclength, Q a fixed point on M , Q = c(0). Then the function t → l Q (t) = |Qc(t)| has the right derivative and where ϕ ′ c (t) is the angle between c(t) and the geodesic arc connecting c(0) to Q. We choose the parametrization (4.4) Figure 2.
Taking into account that P F is the w. F-T point of A • B • C • D • , we derive the (right) derivative of (4.2) with respect to the variable l A • and considering (4.4), we get Similarly, let the point D • , the length parameterized curve c(t) and the point P F that belongs in c(t) be given in the neighborhood W A . Taking into account (4.3) and (4.4), we obtain Replacing (4.6) -(4.8) in (4.5), we obtain Similarly, working cyclically and differentiating (1.1) with respect to l B • , and choosing the parametrization l B • (t ′ ) = t ′ , differentiating (1.1) with respect to l C • and choosing the parametrization l C • (t ′′ ) = t ′′ and differentiating (1.1) with respect to l D • and choosing the parametrization l D • (t ′′′ ) = t ′′′ , we derive three relations, respectively, The equations (4.9) -(4.12) could be written in a unified form Consider the 4-inverse w. F-T problem under the condition Solving the linear system (4.9) -(4.10) with respect to w A • and w B • , we obtain Similarly, solving the system (4.9) and (4.11) with respect to w A • , w C • , we obtain We write (4.14) in the following form: From the derived equations (4.9) -(4.15) we obtain a balancing condition of four tangent vectors at P F which are located at the tangent plane T PF (M ) at P F having their weighted sum zero. By this approach, we deduce the invariance property of the w. F-T point P F for a given convex quadrilateral A • B • C • D • which states that: Suppose that there is a convex quadrilateral A • B • C • D • on M and a nonnegative weight w R corresponds at each vertex R, for R ∈ {A • B • C • D • }. Assume that the w. F-T point P F point is an interior point of A • B • C • D • . If P F is connected with every vertex R and a point S is selected with a non-negative weight w R such that S belongs to the geodesic that is defined by the geodesic arc P F R, for R ∈ {A • , B • , C • , D • } and S ∈ {A, B, C, D} and the convex quadrilateral ABCD is constructed such that the corresponding weighted Fermat-Torricelli point P ′ F is not a vertex of ABCD, then the w. F-T point P ′ F is identical with P F . Squaring both parts of (4.9) and (4.16) and adding the two derived equations, we get (4.17) Applying the same process and exchanging the indices D • ↔ B • in (4.17), we get (4.18) The invariance property of the w. F-T P F gives us the ability to transform the initial quadrilateral A • B • C • D • to ABCD and applying the inverse of the exponential mapping with respect to P F , we can move to A ′ B ′ C ′ D ′ (see Figs. 2, 3) such that where R belongs to the geodesic P F R • , for R ∈ {A, B, C, D}, R ′ ∈ {A ′ , B ′ , C ′ , D ′ }. Furthermore, we construct the symmetric points of A, C with respect to P F , A ⋆ and C ⋆ , respectively, such that l A ⋆ (P F ) = l A (P F ) and l C ⋆ (P F ) = l C (P F ) on T PF (M ). Similarly,we construct the symmetric points of A ′ , C ′ with respect to P F , A ′⋆ and C ′⋆ , respectively, such that l A ′⋆ (P F ) = l A ′ (P F ) and l C ′⋆ (P F ) = l C ′ (P F ) on T PF (M ) (see Figs. 2 and 3). Taking into consideration Fig. 3, we deduce Proof of Proposition 3: Applying the invariance property (geometric plasticity) of P F we transform the initial quadrilateral A • B • C • D • to ABCD and applying the inverse of the exponential mapping with respect to P F , we get A ′ B ′ C ′ D ′ such that |P F R| = | exp −1 PF (R)| = |P F R ′ | = w R , for R ∈ {A, B, C, D} and R ′ ∈ {A ′ , B ′ , C ′ , D ′ }. We conclude that A ′ B ′ C ′ D ′ is a parallelogram because the diagonals bisect.
We proceed by proving the following theorem: Theorem 3. The geometrization of the plasticity of convex quadrilaterals on M is given by the following two classes of parallelograms: Class A: Parallelograms derived from the initial quadrilateral A • B • C • D • to the tangent plane at P F by taking the symmetric points of A ′ and C ′ with respect to P F such that |P F R| = | exp −1 PF (R)| = |P F R ′ | = w R , where R lies on the geodesic P F R • for R ∈ {A, B, C, D} and R ′ ∈ {A ′ , B ′ , C ′ , D ′ }.
Class B: Parallelograms derived from the initial quadrilateral to the tangent plane at P F by taking the symmetric points of B ′ and D ′ with respect to P F .

Proof of Theorem 3:
We consider the following two cases: Applying Theorem 2, we derive class A of Theorem 3. 2 and applying Theorem 2, we derive class B of Theorem 3.
We note that class A and class B contain similar and non similar parallelograms. Similar parallelograms are derived by the condition of the 4-inverse w. F-T problem:  We notice that an increase of w D • from 0.4 to 0.5 causes a decrease to w A • , w C • and an increase to w B • (plasticity). The weights that correspond to Fig. 4 derive a parallelogram A ′⋆ B ′ C ′⋆ D ′ that belongs to class A and the weights that correspond to Remark 4. We have excluded some trivial cases where the weighted quadrilaterals can be transformed directly to parallelograms and we consider quadrilaterals that the initial starting values of their weights which satisfy the plasticity equations must also satisfy the inequalities The author is grateful to Professor Dr.Vladimir Rovenski for many useful discussions, comments on the paper and for communicating to him remark 2.