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Axioms
, Volume 12 (5) – Apr 27, 2023

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axioms Article On an Indeﬁnite Metric on a Four-Dimensional Riemannian Manifold 1 1 2, Dimitar Razpopov , Georgi Dzhelepov and Iva Dokuzova * Department of Mathematics and Informatics, Faculty of Economics, Agricultural University of Plovdiv, 12 Mendeleev Blvd., 4000 Plovdiv, Bulgaria; razpopov@au-plovdiv.bg (D.R.); dzhelepov@abv.bg (G.D.) Department of Algebra and Geometry, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria * Correspondence: dokuzova@uni-plovdiv.bg Abstract: Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive deﬁnite metric, the manifold is endowed with an additional tensor structure of type (1, 1), whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures deﬁne an indeﬁnite metric. With the help of the indeﬁnite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we deﬁne a vector force ﬁeld. Further, we obtain the circulation of the vector force ﬁeld along the curve, as well as the ﬂux of the curl of this vector force ﬁeld across the curve. Finally, we ﬁnd a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space. Keywords: Riemannian manifold; indeﬁnite metric tensor; length; area; Green’s formula MSC: 53B30; 53A04; 26B15; 26B20 1. Introduction Citation: Razpopov, D.; Dzhelepov, Riemannian manifolds with additional tensor structures are extensively studied in G.; Dokuzova, I. On an Indeﬁnite modern differential geometry. Riemannian almost product manifolds and almost Hermitian Metric on a Four-Dimensional Riemannian Manifold. Axioms 2023, manifolds are examples of such manifolds. Four-dimensional Riemannian manifolds with 12, 432. https://doi.org/10.3390/ circulant structures are associated with Riemannian almost product manifolds [1], and axioms12050432 four-dimensional Riemannian manifolds with skew-circulant structures are associated with Hermitian manifolds [2]. Academic Editors: Zhigang Wang, In this paper, we continue our research, previously addressed in [2,3], on Riemannian Yanlin Li, Juan De Dios Pérez and manifolds with an additional tensor structure, whose fourth power is minus the identity. Emil Saucan The additional structure is skew-circulant, i.e., its components form a skew-circulant matrix. Received: 11 April 2023 The properties and some applications of such matrices can be found in [4–9]. We deﬁne an Accepted: 25 April 2023 indeﬁnite metric on the manifold using the Riemannian metric and the additional structure, Published: 27 April 2023 and obtain some useful formulae with respect to this metric, which are analogs of well- known formulae (such as length and area of a circle, circulation of a vector force ﬁeld, and Green’s formula) in the Euclidean case. We can ﬁnd more motivations for our work from several papers (see [10–12]). Copyright: © 2023 by the authors. If k is a simple closed curve in a plane, then it surrounds some region in the plane. Licensee MDPI, Basel, Switzerland. Green’s theorem transforms the line integral around k into a double integral over the region This article is an open access article inside k. In physics, this provides the relationship between the circulation C = F.ds of distributed under the terms and the vector force ﬁeld F around the path k and the ﬂux, done by the curl of F, across the conditions of the Creative Commons region inside k. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Axioms 2023, 12, 432. https://doi.org/10.3390/axioms12050432 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 432 2 of 13 Green’s theorem is a special case of Stokes’ theorem. Both theorems are widely used in the study of electric and magnetic ﬁelds. The modern approach to these theorems on manifolds using differential forms is exhibited, for example, in [13–18]. We consider a four-dimensional Riemannian manifold M with an additional tensor ﬁeld S of type (1, 1), whose fourth power is minus the identity. The structure S is compatible with the metric g, such that an isometry is induced in every tangent space T M on M. Both structures, g and S, deﬁne an indeﬁnite metric g ˜ [2]. The metric g ˜ determines space-like, isotropic and time-like vectors in T M. We consider circles k , with respect to g ˜, in special p i two-planes b of T M, constructed on space-like or time-like vectors.We calculate their lengths and areas (with respect to g ˜), which, in some cases, are imaginary or negative numbers. We note that some problems related to circles, concerning their lengths or areas considered in terms of indeﬁnite metrics, are addressed in [19–23]. Finally, we obtain analogs of Green’s theorem that provide a relation between the circulation of the vector force ﬁeld F around a closed curve (in particular, a circle) k in b and the ﬂux, done by the i i curl of F, across the region inside k . The paper is organized as follows. In Section 2, we provide some facts, deﬁnitions and statements, which are necessary for the present considerations. In Section 3, we introduce a special two-plane b of T M and determine an equation of a circle k in b with respect 1 1 1 to g ˜. In Sections 3.1 and 3.2 we calculate the length and area of k . In Section 3.3, we ﬁnd the circulation of a vector force ﬁeld F around the smooth closed curve, k , and the ﬂux, done by the curl of F, across the region inside k . In Section 4, we introduce a two-plane, b , of T M and determine an equation of a circle k in b with respect to g ˜. Further, we 2 p 2 2 calculate the length and area of k . We derive the circulation of a vector force ﬁeld F around a smooth closed curve k and the ﬂux, done by the curl of F, across the region inside k . All 2 2 values obtained in Sections 3 and 4 are calculated with respect to g ˜. 2. Preliminaries In this paper, we study a four-dimensional Riemannian manifold, equipped with tensor structures whose component matrices are right skew-circulant. Thus, we recall the deﬁnition of such matrices. They are Toeplitz matrices, and were addressed in [4,6]. The real right skew-circulant matrix with the ﬁrst row (a , a , a , a ) 2 R is a square 1 2 3 4 matrix of the form: 0 1 a a a a 1 2 3 4 B C a a a a 4 1 2 3 B C @ A a a a a 3 4 1 2 a a a a 2 3 4 1 The skew-circulant matrices form a vector space with the following basis: 0 1 0 1 1 0 0 0 0 1 0 0 B C B C 0 1 0 0 0 0 1 0 B C B C E = , E = , @ A @ A 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 B C B C 0 0 0 1 1 0 0 0 B C B C E = , E = . 3 4 @ A @ A 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 Further, we use the matrix S = E . We note that E is the identity matrix and S = E , 2 3 S = E . These matrices are also a part of our considerations. The square of S gives J = E , 4 3 which is an example of the well-known complex structure. We equip a four-dimensional differentiable manifold M with a tensor structure S of type (1, 1), such that S satisﬁes S = id. (1) Axioms 2023, 12, 432 3 of 13 We suppose that, at each point on M, the component matrix of S, with respect to a basis in the tangent space T M, is skew-circulant. The matrix E is one solution of the equation S = E. Then, we consider the skew- circulant structure S whose component matrix, with respect to a basis in T M, is 0 1 0 1 0 0 B C 0 0 1 0 B C S = . (2) @ A 0 0 0 1 1 0 0 0 Obviously, S satisﬁes (1) and J = S is a complex structure. Let g be a positive deﬁnite metric on M, which is compatible with S, i.e., g(Su, Sv) = g(u, v). (3) Here, and anywhere in this work, u, v, w, e , e stand for arbitrary smooth vector ﬁelds 1 2 on M or arbitrary vectors in the tangent space T M, p 2 M. The conditions (2) and (3) imply that the matrix of g, with respect to the same basis, has the following form: 0 1 A B 0 B B C B A B 0 B C (g ) = , i j @ A 0 B A B B 0 B A where A = A( p) and B = B( p) are smooth functions of an arbitrary point p on M. Moreover, A( p) > 2B( p) > 0 in order for g to be positive deﬁnite. Such a manifold ( M, g, S) was introduced in [2]. If u is a nonzero vector on ( M, g, S), then, according to (2), we have Su 6= u. Thus, 2 3 the angle j between u and Su belongs to the interval (0, p). The vectors u, Su, S u and S u determine six angles, which satisfy equalities [2]: 2 2 3 \(u, Su) = \(Su, S u) = \(S u, S u) = j, 3 2 3 \(S u, u) = p j, \(u, S u) = \(Su, S u) = . (4) 3 2 Deﬁnition 1. A basis of type fS u, S u, Su, ug of T M is called an S-basis. In this case, we say that the vector u induces an S-basis of T M. In [2], the conditions under which such a basis exists are described, as well as the following statement: 3 2 Lemma 1. Let a vector u induce an S-basisfS u, S u, Su, ug in T M. Then, the angle j = \(u, Su) satisﬁes inequalities p 3p < j < . (5) 4 4 The associated metric g ˜ on ( M, g, S) is determined by g ˜(u, v) = g(u, Sv) + g(Su, v). (6) The matrix of its components is 0 1 2B A 0 A B C A 2B A 0 B C (g ˜ ) = . i j @ A 0 A 2B A A 0 A 2B Axioms 2023, 12, 432 4 of 13 Two of the eigenvalues of g ˜ are negative, and the other two are positive. So, g ˜ has i j signature (2, 2) and it is an indeﬁnite metric [2]. According to (6), for an arbitrary vector v the following is valid: 2 2 g(v, v) = 2g(v, Sv) = R , R 2 R. (7) The norm of every vector u and the cosine of j are given by the following equalities: g(u, Su) kuk = g(u, u), cos j = . (8) g(u, u) In the rest of the paper, we assume that kuk = 1 and, using (8), we obtain cos j = g(u, Su). (9) 3 2 Due to (3), (6), (8) and (9) we state that the normalized S-basisfS u, S u, Su, ug satisﬁes the following equalities: 2 2 3 3 g ˜(u, u) = g ˜(Su, Su) = g ˜(S u, S u) = g ˜(S u, S u) = 2 cos j, 2 2 3 3 g ˜(u, Su) = g ˜(Su, S u) = g ˜(S u, S u) = g ˜(S u, u) = 1. (10) 2 3 g ˜(u, S u) = g ˜(Su, S u) = 0. A circle k in a two-plane of T M of a radius R centered at the origin p 2 T M, with p p respect to the associated metric g ˜ on ( M, g, S), is determined by (7), where v is the radius vector of an arbitrary point on k. Further, we consider circles k and k , and the regions D and D inside them, in 1 2 1 2 two different subspaces b and b of T M, spanned by two-planes fu, S ug and fu, Sug, 1 2 p 2 3 respectively. According to (4), (5) and (10), the vectors u, Su, S u and S u are space-like p p p 3p if, and only if, j 2 , , and they are time-like if, and only if, j 2 ( , ). Therefore, 4 2 2 4 p p the two-planes b and b are space-like when j 2 , , and they are time-like when 1 2 4 2 p 3p j 2 ( , ). 2 4 3 2 3 2 3 Remark 1. The rest of the two-planes fu, S ug, fSu, S ug, fSu, S ug, fS u, S ug constructed 3 2 with the basis vectors of fS u, S u, Su, ug in T M have the same properties as b or b . p 2 We note that the two-plane b also belongs to the tangent space of the associated Hermitian manifold ( M, g, J = S ) with the complex structure J. 3. Circles in the Two-Plane b Due to (4), it is true that the vectors u and S u form an orthonormal basis of b . The coordinate system p on b , wherein u is on the axis p and S u is on the axis p , is an xy 1 x y orthonormal coordinate system of b . A circle k in b centered at the origin p, with respect to g ˜ on ( M, g, S), is deﬁned 1 1 by (7). The equation of k with respect to p is obtained as follows: xy Theorem 1 ([3]). Let g ˜ be the associated metric on ( M, g, S) and let b be a two-plane in T M 1 p 2 2 with a basis fu, S ug. If p is a coordinate system such that u 2 p , S u 2 p , then, the equation xy x y of the circle (7) in b is given by: 2 2 2 2 cos jx + 2 cos jy = R . (11) The only closed curve k , determined by (11), is a circle in terms of g, with the following parameters: p p Case (A) j 2 , and R > 0; 4 2 p 3p 2 Case (B) j 2 , and R < 0. 2 4 Axioms 2023, 12, 432 5 of 13 The two-plane b is constructed on space-like vectors in Case (A), and b is constructed 1 1 on time-like vectors in Case (B). 3.1. Length of a Circle with Respect to g ˜ Firstly, we consider Case (A). The circle (7) has a radius R > 0 and the angle j satisﬁes p p < j < . (12) 4 2 Theorem 2. The circle k with (12) and a radius R > 0 has length L = 2p R. (13) Proof. Let v = xu + yS u be a radius vector of an arbitrary point on the circle k . Then, dv = dxu + dyS u is a tangent vector on k . The length L of k with respect to g ˜ is 1 1 determined, as usual, by: dL = g ˜(dv, dv). Then, using (10) and (11), we obtain: I q 2 2 L = 2 cos jdx + 2 cos jdy . (14) We substitute R R x = p cos t, y = p sin t, t 2 [0, 2p], 2 cos j 2 cos j into (14) and ﬁnd (13). Now, we consider Case (B). The circle k has a radius R = ri, r > 0, i = 1 and the angle j satisﬁes p 3p < j < . (15) 2 4 Therefore, the Equation (11) transforms into 2 2 2 2 cos jx + 2 cos jy = r . (16) By calculations similar to those of Case (A) we ﬁnd the integral (14) over k with (16). Using the substitutions r r x = p cos t, y = p sin t, t 2 [0, 2p], 2 cos j 2 cos j we obtain Proposition 1. The circle k with (15) and a radius R = ri, r > 0, i = 1 has an imagi- nary length L = 2p R. 3.2. Area of a Circle with Respect to g ˜ For Case (A) we state the following: Theorem 3. The area A of the circle k with (12) with a radius R > 0 is 00 2 A = p R . (17) Axioms 2023, 12, 432 6 of 13 2 2 Proof. We denote, by cos f \(u, S u) and sin\(u, S u), the cosine and the sine of the angle 2 2 \(u, S u) with respect to g ˜. Considering g ˜(u, S u) = 0 (presented in (10)), we have cos f \(u, S u) = 0, and, hence, sin\(u, S u) = 1. (18) In the coordinate plane p , we construct a parallelogram with locus vectors dxu and xy dySu. For its area A with respect to g ˜ we obtain q q 00 2 d A = g ˜(dxu, dxu) g ˜(dySu, dySu)sin\(u, S u). We apply (10) and (18) in the latter equality and ﬁnd d A = 2 cos jdxdy. (19) We integrate (19) over the region D inside k and calculate 1 1 Z Z A = 2 cos j dxdy, (20) with 2 2 2 D : 2 cos jx + 2 cos jy R . We substitute R R p p x = r cos t, y = r sin t, t 2 [0, 2p], r 2 [0, 1], 2 cos j 2 cos j and Jacobian 4 = r into the integral (20) and obtain (17). 2 cos j Now, we consider Case (B). 00 2 Proposition 2. The area A of the circle k with (15) and a radius R = ri, r > 0, i = 1 has a negative value 00 2 A = p R . (21) Proof. The circle k has an Equation (16) with conditions (15) and a radius R = ri, where r > 0, i = 1. By calculations similar to those of Case (A), we ﬁnd that the area of k is given by Z Z A = 2 cos j dxdy, (22) with 2 2 2 D : 2 cos jx 2 cos jy r . We substitute r r p p x = r cos t, y = r sin t, t 2 [0, 2p], r 2 [0, 1], 2 cos j 2 cos j 00 2 and Jacobian 4 = r into the integral (22) and obtain A = pr , which im- 2 cos j plies (21). 3.3. Circulation and Flux with Respect to g ˜ We consider a closed curve k in b , given by 1 1 x = x(t), y = y(t), t 2 [t , t ], (23) 1 2 Axioms 2023, 12, 432 7 of 13 where x(t ) = x(t ), y(t ) = y(t ). 1 2 1 2 Let F(x, y) = P(x, y)u + Q(x, y)S u (24) be a vector force ﬁeld on the curve k . For the circulation C of a vector ﬁeld F along a curve k we assume the following deﬁnition: C = g ˜(F, dv), (25) where v is the radius vector of a point on k. We denote, by D , the region inside k . For both cases (A) and (B) of circle (11) the 1 1 following statements hold. Theorem 4. The circulation C, done by the force (24) along the curve (23), is expressed by C = 2 cos j (P(x, y)dx + Q(x, y)dy), (26) p p p 3p where j 2 , [ , . 4 2 2 4 Proof. Let v = xu + yS u be the radius vector of a point on k . By virtue of (10) and (24), and bearing in mind dv = dxu + dyS u, we obtain g ˜(F, dv) = 2 cos j P(x, y)dx + Q(x, y)dy . (27) ( ) Obviously, (26) follows from (23), (25) and (27). We determine a vector w in T M by the equality w = p cos ju Su + cos jS u , (28) 1 2 cos j p p p 3p where j 2 , [ , . By using (1), (3) and (9) it is easy to verify that 4 2 2 4 g(w, u) = g(w, S u) = 0, g(w, w) = 1. We construct an orthonormal coordinate system Oxyz, such that u 2 Ox, S u 2 Oy, w 2 Oz. We suppose that the curl of F, determined by (24), with respect to Oxyz, is curl F = (Q P )w. x y The ﬂux T of the vector ﬁeld curl F across the region D inside the curve k is given by 1 1 Z Z T = g ˜(curl F, w)d A . (29) With the help of (10) and (28) we obtain g ˜(w, w) = 2 cos j. Then, from (19) and (29) we state the following. Theorem 5. The ﬂux T of the vector ﬁeld curl F across the region D inside the curve (23) is expressed by Z Z T = 4 cos j (Q P )dxdy, (30) x y p p p 3p where j 2 , [ , . 4 2 2 4 Axioms 2023, 12, 432 8 of 13 On the other hand, due to Green’s formula, we have Z Z I (Q P )dxdy = (Pdx + Qdy). x y D k Bearing in mind the above formula we obtain the statements that follow. Theorem 6. The relation between the circulation (26) and the ﬂux (30) is determined by T = 2 cos jC. Corollary 1. The relation between the circulation C and the ﬂux T is (a) T = C, in case j = ; 2p (b) T = C, in case j = . 4. Circles in the Two-Plane b Lemma 2 ([3]). Let b be the 2-plane spanned by unit vectors u and Su. The system of vectors fe , e g, determined by the equalities 1 2 1 1 e = p (u + Su), e = p (u + Su), (31) 1 2 2(1 + cos j) 2(1 cos j) is an orthonormal basis of b with respect to g. The coordinate system p on b , such that e is on the axis p and e is on the xy 2 1 x 2 axis p , is orthonormal. Due to (10), we ascertain that the system fe , e g satisﬁes the 1 2 following equalities: 2 cos j + 1 2 cos j 1 ˜ ˜ ˜ g(e , e ) = , g(e , e ) = , g(e , e ) = 0. (32) 1 1 2 2 1 2 1 + cos j 1 cos j A circle k in b centered at the origin p, with respect to g ˜ on ( M, g, S), is deﬁned 2 2 by (7). The equation of k with respect to p is obtained as below. xy Theorem 7 ([3]). Let g ˜ be the associated metric on ( M, g, S) and let b = fu, Sug be a 2-plane in T M with an orthonormal basis (31). If p is a coordinate system, such that e 2 p , e 2 p , p xy x y 1 2 then the equation of the circle (7) in b is given by 2 cos j + 1 2 cos j 1 2 2 2 x + y = R . (33) 1 + cos j 1 cos j The only closed curve k , determined by (33), is an ellipse in terms of g with parameters: p p 2 Case (A) j 2 , and R > 0; 4 3 2p 3p 2 Case (B) j 2 , and R < 0. 3 4 The two-plane b is constructed on space-like vectors in Case (A), and b is constructed 2 2 on time-like vectors in Case (B). 4.1. Length of a Circle with Respect to g ˜ Firstly, we consider Case (A). The circle (7) has a radius R > 0 and j satisﬁes p p < j < . (34) 4 3 Theorem 8. The circle k with (34) and a radius R > 0 has length L = 2p R. (35) Axioms 2023, 12, 432 9 of 13 Proof. The radius vector v of an arbitrary point on the curve k is v = xe + ye . Then, 2 1 2 dv = dxe + dye is a tangent vector on k . The length L of k , with respect to g ˜, is 1 2 2 2 dL = g ˜(dv, dv). From (32), we ﬁnd 2 cos j + 1 2 cos j 1 2 2 g ˜(dv, dv) = dx + dy . 1 + cos j 1 cos j Then, we obtain 2 cos j + 1 2 cos j 1 2 2 L = dx + dy . (36) 1 + cos j 1 cos j We substitute s s 1 + cos j 1 cos j x = R cos t, y = R sin t, t 2 [0, 2p] 2 cos j + 1 2 cos j 1 into (36) and obtain 2p 2 2 2 L = R sin t + R cos tdt, which implies (35). Now, we consider Case (B). The circle k has a radius R = ri, r > 0, i = 1 and the angle j satisﬁes 2p 3p < j < . (37) 3 4 The Equation (33) transforms into 2 cos j + 1 2 cos j 1 2 2 2 x + y = r . (38) 1 + cos j 1 cos j By calculations similar to those of Case (A), we ﬁnd the integral (36) over k with (38). Using the substitutions s s 1 + cos j 1 cos j x = r cos t, y = r sin t, t 2 [0, 2p], 2 cos j + 1 2 cos j 1 we calculate the length of k and formulate the next proposition. Proposition 3. The circle k with (37) and a radius R = ri, r > 0, i = 1 has an imagi- nary length L = 2p R. 4.2. Area of a Circle with Respect to g For Case (A) we state the following. Theorem 9. The area A of the circle k with (34) and a radius R > 0 is 00 2 A = p R . (39) Axioms 2023, 12, 432 10 of 13 Proof. Let us denote, by cos f q, the cosine of q = \(e , e ) with respect to g ˜, which is 1 2 g ˜(e , e ) 1 2 cos f q = p p . g ˜(e , e ) g ˜(e , e ) 1 1 2 2 Then, using (32), we derive cos f q = 0, which implies sinq = 1. (40) In the coordinate plane p , we construct a parallelogram with locus vectors dxe and xy 1 dye . For its area A with respect to g ˜ we obtain q q d A = g ˜(dxe , dxe ) g ˜(dye , dye )sinq. 1 1 2 2 We apply (32) and (40) in the above equality and obtain 4 cos j 1 d A = dxdy. (41) sin j We integrate (41) over the region D inside k and calculate 2 2 Z Z 4 cos j 1 A = dxdy, (42) sin j with 2 cos j + 1 2 cos j 1 2 2 2 D : x + y R . 1 + cos j 1 cos j We substitute s s 1 + cos j 1 cos j x = Rr cos t, y = Rr sin t, t 2 [0, 2p], r 2 [0, 1], 2 cos j + 1 2 cos j 1 sin j and Jacobian 4 = R r into (42). Finally we get (39). 4 cos j 1 Now, we consider Case (B). 00 2 Proposition 4. The area A of the circle k with (34) and a radius R = ri, r > 0, i = 1 has a negative value 00 2 A = p R . (43) Proof. The circle k has Equation (38), with conditions (37) and a radius R = ri, where r > 0, i = 1. By calculations analogous to the previous case, we ﬁnd that the area of k is given by Z Z 4 cos j 1 A = dxdy, (44) sin j with 2 cos j + 1 2 cos j 1 2 2 2 D : x y r . 1 + cos j 1 cos j We substitute s s 1 + cos j 1 cos j x = rr cos t, y = rr sin t, t 2 [0, 2p], r 2 [0, 1], 2 cos j + 1 2 cos j 1 Axioms 2023, 12, 432 11 of 13 sin j 2 00 2 and Jacobian 4 = p r r into the integral (44) and obtain A = pr , which 4 cos j 1 implies (43). 4.3. Circulation and Flux with Respect to g ˜ We consider a closed curve k in b , given by 2 2 x = x(t), y = y(t), t 2 [t , t ], (45) 1 2 where x(t ) = x(t ), y(t ) = y(t ). 1 2 1 2 Let F(x, y) = P(x, y)e + Q(x, y)e (46) 1 2 be a vector force ﬁeld on the curve k . We denote, by D , the region inside k . For both cases (A) and (B) of ellipse (33) the 2 2 following statements hold. Theorem 10. The circulation C of the force (46) along the curve (45) is expressed by 2 cos j + 1 2 cos j 1 C = P(x, y)dx + Q(x, y)dy , (47) 1 + cos j 1 cos j p p 2p 3p where j 2 , [ , . 3 4 3 4 Proof. For the circulation C of a vector force ﬁeld F acting along the curve (45) we use (25), where v = xe + ye is the radius vector of a point on k . Therefore, we have 1 2 2 C = g(F, dv), (48) with a tangent vector dv = dxe + dye on k . Then, by virtue of (32) and (46), we obtain 2 2 2 cos j + 1 2 cos j 1 g ˜(F, dv) = P(x, y)dx + Q(x, y)dy . (49) 1 + cos j 1 cos j Hence, (45), (48) and (49) imply (47). Theorem 11. The ﬂux T of the vector ﬁeld curl F across the region D inside the curve (45) is expressed by q Z Z T = 2 cot j 4 cos j 1 (Q P )dxdy, (50) x y p p 2p 3p where j 2 , [ , . 3 4 3 4 Proof. We determine a vector w in T M by the equality 2 2 2 w = p cos ju (cos j)Su + sin jS u , (51) sin j 1 2 cos j p p 2p 3p where j 2 , [ , . Then, using (1), (3), (9) and (31), we verify that 3 4 3 4 g(w, e ) = g(w, e ) = 0, g(w, w) = 1. 1 2 The coordinate system Oxyz, such that e 2 Ox, e 2 Oy, w 2 Oz, is orthonormal. 1 2 We obtain the curl of F, determined by (46), by the equality curl F = (Q P )w. x y For the ﬂux T of the vector ﬁeld curl F across the region D inside the curve (45) we have Z Z T = g ˜(curl F, w)d A . (52) 2 Axioms 2023, 12, 432 12 of 13 Now, with the help of (32) and (51), we calculate 2 cos j g ˜(w, w) = . sin j Then, from (41) and (52), it follows (50). We introduce the following notations: I I 2 cos j + 1 2 cos j 1 c = Pdx, c = Qdy. (53) 1 2 1 + cos j 1 cos j k k 2 2 On the other hand, due to Green’s formula, we have Z Z I Z Z I P dxdy = Pdx, Q dxdy = Qdy. y x D k D k Bearing in mind the latter equalities we derive the next statement. Theorem 12. The relation between the circulation (47) and the ﬂux (50) is determined by s s 2 cos j 1 2 cos j + 1 T = 2 cot j (1 + cos j) c + (1 cos j) c , 1 2 2 cos j + 1 2 cos j 1 where c and c are given in (53). 1 2 5. Conclusions In this paper, we investigated the properties of special two-planes b and b of T M 1 2 p of a four-dimensional Riemannian manifold ( M, g, S), equipped with an additional in- deﬁnite metric g ˜(,) = g(, S) + g(S,). In these two-planes, circles, with respect to g ˜, are transformed into closed curves in terms of g. Therefore, we can consider analogs of well-known formulae, such as circulation of a vector force ﬁeld along the curve and ﬂux of the curl of a vector force ﬁeld across the curve. It turns out that the length and area, calculated with respect to the indeﬁnite metric, of the circles in b and b are the same as in 1 2 the Euclidean space. Author Contributions: Conceptualization, D.R., G.D. and I.D.; methodology, D.R., G.D. and I.D.; inves- tigation, D.R., G.D. and I.D.; writing—original draft preparation, D.R., G.D. and I.D.; writing—review and editing, D.R., G.D. and I.D.; funding acquisition, D.R. and G.D. All authors have read and agreed to the published version of the manuscript. Funding: This research was partially funded by project 17-12 ”Support for publishing activities”, Agricultural University of Plovdiv, Bulgaria. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: We are grateful to the editors and reviewers for their helpful comments. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Dokuzova, I. Curvature properties of 4-dimensional Riemannian manifolds with a circulant structure. J. Geom. 2017, 108, 517–527. [CrossRef] 2. Dokuzova, I.; Razpopov, D. 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Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 27, 2023

**Keywords: **Riemannian manifold; indefinite metric tensor; length; area; Green’s formula

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