On an Indefinite Metric on a 4-Dimensional Riemannian Manifold

1. Introduction

If k is a simple closed curve in a plane, then it surrounds some region in the plane. Green’s theorem transforms the line integral around k into a double integral over the region inside k. In physics, it gives a relationship between the circulation $C={\oint }_{k}F.\mathrm{d}s$ of the vector force field F around the path k and the flux, done by $curlF$, across the region inside k.

Green’s theorem is a special case of Stokes’ theorem. Both theorems are widely used during the study of electric and magnetic fields. The modern approach to these theorems on manifolds using differential forms is exhibited, for example, in [2,3,9,12,13]. A theorem similar to the theorem of Green, in a special 2-plane of the tangent space on a 3-dimensional Riemannian manifold with circulant structures, is obtained in [5].

We consider a 4-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 $T_{p}M$ on M. Both structures g and S define an indefinite metric $\tilde{g}$[4]. The metric $\tilde{g}$ determines space-like, isotropic and time-like vectors in $T_{p}M$. In special 2-planes ${\beta }_i$ of $T_{p}M$, constructed on space-like and time-like vectors, we consider circles $k_i$ with respect to $\tilde{g}$. We calculate their length and area (with respect to $\tilde{g}$), which in some cases are imaginary or negative numbers. It turns out that these measures are the same as in the Euclidean space. We note that some problems related to circles, their length or area, considered in terms of indefinite metrics, are given in [1,7,8,10,11]. Finally, we obtain analogues of Green’s theorem that give a relation between circulation of the vector force field F around a closed curve (in particular a circle) $k_i$ in ${\beta }_i$ and the flux, done by the curl of F, across the region inside $k_i$.

The paper is organized as follows. In Section 2, we give some facts, definitions and statements, which are necessary for the present considerations. Some of them are obtained in [4,6] and [14]. In Section 3, we introduce a special 2-plane ${\beta }_1$ of $T_{p}M$ and determine an equation of a circle $k_1$ with respect to $\tilde{g}$. In SubSection 3.1 and Section 3.2 we calculate the length and the area of $k_1$. In SubSection 3.3, we find the circulation of a vector force field F around smooth closed curve $k_1$ and the flux, done by the curl of F, across the region inside $k_1$. In Section 4, we introduce a 2-plane ${\beta }_2$ of $T_{p}M$ and determine an equation of a circle $k_2$ with respect to $\tilde{g}$. Further we calculate the length and the area of $k_2$. We get the circulation of a vector force field F around a smooth closed curve $k_2$ and the flux, done by the curl of F, across the region inside $k_2$. We find a relationship between the circulation and the flux in both cases. All values obtained in Section 3 and Section 4 are calculated with respect to $\tilde{g}$.

2. Preliminaries

We consider a 4-dimensional Riemannian manifold M with an additional tensor structure S of type $(1,1)$. In a local coordinate system $(x^1,x^2,x^3,x^4)$ the coordinates of S form the following circulant matrix:

$$S=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ -1& 0& 0& 0\end{array}\right).$$

Thus we have

$$S^4=-\mathrm{id}.$$

(1)

Let g be a positive definite metric on M, which satisfies the equality

$$g(Su,Sv)=g(u,v),u,v\in \mathfrak{X}M.$$

(2)

Such a manifold $(M,g,S)$ is introduced in [4].

Further $u,v,w,e_1,e_2$ will stand for arbitrary smooth vector fields on M or arbitrary vectors in the tangent space $T_{p}M$, $p\in M$.

Let the vector u induce a basis of type $\{S^{3}u,S^{2}u,Su,u\}$. In [4] it is called an S-basis and the following statements about the angles between the basis vectors are obtained.

(i) The angle $\phi$, determined by $\phi =\angle (u,Su),$ satisfies inequalities

$$\frac{\pi }{4}<\phi <\frac{3\pi }{4}.$$

(ii) For the angles between the basis vectors we have

$$\begin{array}{cc}\hfill & \angle (u,Su)=\angle (Su,S^{2}u)=\angle (S^{2}u,S^{3}u)=\phi ,\\ & \angle (S^{3}u,u)=\pi -\phi ,\angle (u,S^{2}u)=\angle (Su,S^{3}u)=\frac{\pi }{2}.\end{array}$$

(3)

The associated metric $\tilde{g}$ on $(M,g,S)$, determined by

$$\tilde{g}(u,v)=g(u,Sv)+g(Su,v),$$

(4)

is necessary indefinite ([4]). Consequently, for an arbitrary vector v it is valid:

$$\tilde{g}(v,v)=2g(v,Sv)=R^2,R^2\in \mathbb{R}.$$

(5)

The norm of every vector u and the cosine of $\phi$ are given by the following equalities:

$$\parallel u\parallel =\sqrt{g(u,u)},cos\phi =\frac{g(u,Su)}{g(u,u)}.$$

(6)

In rest of this paper, we assume that $\parallel u\parallel =1$ and using (6) we have

$$cos\phi =g(u,Su).$$

(7)

Due to (2), (4), (6) and (7) we state that the normal basis $\{S^{3}u,S^{2}u,Su,u\}$ satisfies the following equalities:

$$\begin{array}{cc}\hfill & \tilde{g}(u,u)=\tilde{g}(Su,Su)=\tilde{g}(S^{2}u,S^{2}u)=\tilde{g}(S^{3}u,S^{3}u)=2cos\phi ,\hfill \\ \hfill & \tilde{g}(u,Su)=\tilde{g}(Su,S^{2}u)=\tilde{g}(S^{2}u,S^{3}u)=-\tilde{g}(S^{3}u,u)=1.\hfill \\ \hfill & \tilde{g}(u,S^{2}u)=\tilde{g}(Su,S^{3}u)=0.\hfill \end{array}$$

(8)

A circle k in a 2-plane of $T_{p}M$ of a radius R centered at the origin $p\in T_{p}M$, with respect to the associated metric $\tilde{g}$ on $(M,g,S)$, is determined by (5), where v is the radius vector of an arbitrary point on k.

Farther, we consider circles $k_1$ and $k_2$, and the regions $D_1$ and $D_2$ inside them, in two different subspaces ${\beta }_1$ and ${\beta }_2$, spanned by 2-planes $\{u,S^{2}u\}$ and $\{u,Su\}$, respectively.

3. Circles in the 2-plane ${\beta }_1$

Because of (3), it is true that the vectors u and $S^{2}u$ form an orthonormal basis of ${\beta }_1$. The coordinate system $p_{xy}$ on ${\beta }_1$, such that u is on the axis $p_x$ and $S^{2}u$ is on the axis $p_y$, is an orthonormal coordinate system of ${\beta }_1$.

A circle $k_1$ in ${\beta }_1$ centered at the origin p, with respect to $\tilde{g}$ on $(M,g,S)$, is defined by (5). The equation of $k_1$ with respect to $p_{xy}$ is obtained by the following

Theorem 1. 

[6] Let $\tilde{g}$ be the associated metric on $(M,g,S)$ and let ${\beta }_1$ be a 2-plane in $T_{p}M$ with a basis $\{u,S^{2}u\}$. If $p_{xy}$ is a coordinate system such that $u\in p_x$, $S^{2}u\in p_y$, then the equation of the circle (5) in ${\beta }_1$ is given by

$$2cos\phi x^2+2cos\phi y^2=R^2.$$

(9)

The curve $k_1$, determined by (9), is a circle in terms of g if:

Case (A)

$\phi \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)$ and $R^2>0$;

Case (B)

$\phi \in \left(\frac{\pi }{2},\frac{3\pi }{4}\right)$ and $R^2<0$.

3.1. Length of a circle with respect to $\tilde{g}$

Firstly, we consider Case (A). The circle (5) has a radius $R>0$ and $\phi$ satisfies

$$\frac{\pi }{4}<\phi <\frac{\pi }{2}.$$

(10)

Theorem 2. 

The circle $k_1$ with (10) and a radius $R>0$ has a length

$$L=2\pi R.$$

(11)

Proof. 

Let $v=xu+y{S}^{2}u$ be a radius vector of an arbitrary point on the circle $k_1$. Then $\mathrm{d}v=\mathrm{d}xu+\mathrm{d}y{S}^{2}u$ is a tangent vector on $k_1$. The length L of $k_1$ with respect to $\tilde{g}$ is determined as usual by

$$\mathrm{d}L=\sqrt{\tilde{g}(\mathrm{d}v,\mathrm{d}v)}.$$

Then, using (8) and (9), we obtain

$$L={\oint }_{k_1}\sqrt{2cos\phi \mathrm{d}{x}^2+2cos\phi \mathrm{d}{y}^2}.$$

(12)

We substitute

$$x=\frac{R}{\sqrt{2cos\phi }}cost,y=\frac{R}{\sqrt{2cos\phi }}sint,t\in [0,2\pi ],$$

into (12) and find (11). □

Now, we consider Case (B). The circle $k_1$ has a radius $R=ri,r>0,i^2=-1$ and $\phi$ satisfies

$$\frac{\pi }{2}<\phi <\frac{3\pi }{4}.$$

(13)

Therefore, the equation (9) transforms into

$$2cos\phi x^2+2cos\phi y^2=-r^2.$$

(14)

By calculations similar to Case (A), but considering that R is an imaginary number, we obtain that the circle $k_1$ with (13) has an imaginary length (11).

3.2. Area of a circle with respect $\tilde{g}$

For Case (A) we state the following

Theorem 3. 

The area $\mathtt{A}$ of the circle $k_1$ with (10) and a radius $R>0$ is

$$\mathtt{A}=\pi R^2.$$

(15)

Proof. 

We denote by $\widetilde{cos}\angle (u,S^{2}u)$ and $\widetilde{sin}\angle (u,S^{2}u)$ the cosine and the sine of the angle $\angle (u,S^{2}u)$ with respect to $\tilde{g}$. Considering $\tilde{g}(u,S^{2}u)=0$ (presented in (8)), we have

$$\widetilde{cos}\angle (u,S^{2}u)=0,$$

and hence

$$\widetilde{sin}\angle (u,S^{2}u)=1.$$

(16)

In the coordinate plane $p_{xy}$, we construct a parallelogram with locus vectors $\mathrm{d}xu$ and $\mathrm{d}ySu$. For its area $\mathtt{A}$ with respect to $\tilde{g}$ we get

$$\mathrm{d}\mathtt{A}=\sqrt{\tilde{g}(\mathrm{d}xu,\mathrm{d}xu)}\sqrt{\tilde{g}(\mathrm{d}ySu,\mathrm{d}ySu)}\widetilde{sin}\angle (u,S^{2}u).$$

We apply (8) and (16) in the latter equality and find

$$\mathrm{d}\mathtt{A}=2cos\phi \mathrm{d}x\mathrm{d}y.$$

(17)

We integrate (17) over the region $D_1$ inside $k_1$ and calculate

$$\mathtt{A}=2cos\phi \int {\int }_{D_1}\mathrm{d}x\mathrm{d}y,$$

(18)

with

$$D_1:2cos\phi x^2+2cos\phi y^2\le R^2.$$

We substitute

$$x=\frac{R}{\sqrt{2cos\phi }}\rho cost,y=\frac{R}{\sqrt{2cos\phi }}\rho sint,t\in [0,2\pi ],\rho \in [0,1],$$

and Jacobian $▵=\frac{R^2}{2cos\phi }\rho$ into the integral (18) and obtain (15). □

Now, we consider Case (B). The circle $k_1$ has an equation (14) with conditions (13) and a radius $R=ri,r>0,i^2=-1$. By calculations similar to Case (A), we find that the area of $k_1$ is given by (15). In this case, $\mathtt{A}$ has a negative value.

3.3. Circulation and flux with respect to the metric $\tilde{g}$

We consider a closed curve $k_1$ in ${\beta }_1$, given by

$$x=x\left(t\right),y=y\left(t\right),t\in [\alpha ,\beta ],$$

(19)

where $x\left(\alpha \right)=x\left(\beta \right)$, $y\left(\alpha \right)=y\left(\beta \right)$.

Let

$$F(x,y)=P(x,y)u+Q(x,y)S^{2}u$$

(20)

be a vector force field on the curve $k_1$.

For the circulation C of a vector field F along a curve k we assume the following definition

$$C={\oint }_k\tilde{g}(F,\mathrm{d}v),$$

(21)

where v is the radius vector of a point on k.

We denote by $D_1$ the region inside $k_1$. For both cases (A) and (B) of the circle (9) the following statements are valid.

Theorem 4. 

The circulation C, done by the force (20) along the curve (19), is expressed by

$$\begin{array}{c}\hfill C=2cos\phi {\oint }_{k_1}(P(x,y)\mathrm{d}x+Q(x,y)\mathrm{d}y),\end{array}$$

(22)

where $\phi \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\frac{3\pi }{4}\right)$.

Proof. 

Let $v=xu+y{S}^{2}u$ be the radius vector of a point on $k_1$. By virtue of (8) and (20), and bearing in mind $\mathrm{d}v=\mathrm{d}xu+\mathrm{d}y{S}^{2}u$, we obtain

$$\begin{array}{c}\hfill \tilde{g}(F,\mathrm{d}v)=2cos\phi \left(P(x,y)\mathrm{d}x+Q(x,y)\mathrm{d}y\right).\end{array}$$

(23)

Evidently (22) follows from (19), (21) and (23). □

We determine a vector w in $T_{p}M$ by the equality

$$w=\frac{1}{\sqrt{1-2{cos}^2\phi }}\left(cos\phi u-Su+cos\phi S^{2}u\right),$$

(24)

where $\phi \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\frac{3\pi }{4}\right)$. By using (1), (2) and (7) it is easy to verify that

$$g(w,u)=g(w,S^{2}u)=0,g(w,w)=1.$$

We construct an orthonormal coordinate system $Oxyz$, such that $u\in Ox$, $S^{2}u\in Oy$, $w\in Oz$.

We suppose that the curl of F, determined by (20), with respect to $Oxyz$ is

$$\mathrm{curl}F=(Q_x-P_y)w.$$

The flux T of the vector field $curlF$ across the region $D_1$ inside the curve $k_1$ is given by

$$T=\int {\int }_{D_1}\tilde{g}(curlF,w)\mathrm{d}\mathtt{A}.$$

(25)

With the help of (8) and (24) we get $\tilde{g}(w,w)=-2cos\phi$. Then from (17) and (25) we state the following

Theorem 5. 

The flux T of the vector field $\mathrm{curl}F$ across the region $D_1$ inside (19) is expressed by

$$T=-4{cos}^2\phi \int {\int }_{D_1}(Q_x-P_y)\mathrm{d}x\mathrm{d}y,$$

(26)

where $\phi \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\frac{3\pi }{4}\right).$

On the other hand, due to Green’s formula, we have

$$\begin{array}{c}\hfill \int {\int }_D(Q_x-P_y)\mathrm{d}x\mathrm{d}y={\oint }_k(P\mathrm{d}x+Q\mathrm{d}y).\end{array}$$

Bearing in mind the above formula we obtain the following statement.

Theorem 6. 

The relation between the circulation (22) and the flux (26) is determined by

$$T=-2cos\phi C.$$

Corollary 1. 

The relation between the circulation C and the flux T is

a)

$T=-C,$ in case $\phi =\frac{\pi }{3}$;

b)

$T=C,$ in case $\phi =\frac{2\pi }{3}$.

4. Circles in the 2-plane ${\beta }_2$

Lemma 1. 

[6] Let ${\beta }_2$ be the 2-plane spanned by unit vectors u and $Su$. The system of vectors $\{e_1,e_2\}$, determined by the equalities

$$e_1=\frac{1}{\sqrt{2(1+cos\phi )}}(u+Su),e_2=\frac{1}{\sqrt{2(1-cos\phi )}}(-u+Su),$$

(27)

is an orthonormal basis of ${\beta }_2$ with respect to g.

The coordinate system $p_{xy}$ on ${\beta }_2$, such that $e_1$ is on the axis $p_x$ and $e_2$ is on the axis $p_y$, is orthonormal. Due to (8) we obtain that the system $\{e_1,e_2\}$ satisfies the following equalities:

$$\tilde{g}(e_1,e_1)=\frac{2cos\phi +1}{1+cos\phi },\tilde{g}(e_2,e_2)=\frac{2cos\phi -1}{1-cos\phi },\tilde{g}(e_1,e_2)=0.$$

(28)

A circle $k_2$ in ${\beta }_2$ centered at the origin p, with respect to $\tilde{g}$ on $(M,g,S)$, is defined by (5). The equation of $k_2$ with respect to $p_{xy}$ is obtained in the following

Theorem 7. 

[6] Let $\tilde{g}$ be the associated metric on $(M,g,S)$ and let ${\beta }_2=\{u,Su\}$ be a 2-plane in $T_{p}M$ with an orthonormal basis (27). If $p_{xy}$ is a coordinate system such that $e_1\in p_x$, $e_2\in p_y$, then the equation of the circle (5) in ${\beta }_2$ is given by

$$\frac{2cos\phi +1}{1+cos\phi }{x}^2+\frac{2cos\phi -1}{1-cos\phi }{y}^2=R^2.$$

(29)

The curve $k_2$, determined by (29), is an ellipse in terms of g if:

Case (A)

$\phi \in \left(\frac{\pi }{4},\frac{\pi }{3}\right)$ and $R^2>0$;

Case (B)

$\phi \in \left(\frac{2\pi }{3},\frac{3\pi }{4}\right)$ and $R^2<0$.

Firstly, we consider Case (A). The circle (5) has a radius $R>0$ and $\phi$ satisfies

$$\frac{\pi }{4}<\phi <\frac{\pi }{3}.$$

(30)

Theorem 8. 

The circle $k_2$ with (30) and a radius $R>0$ has a length

$$L=2\pi R.$$

(31)

Proof. 

The radius vector v of an arbitrary point on the curve $k_2$ is $v=x{e}_1+y{e}_2.$ Then $\mathrm{d}v=\mathrm{d}x{e}_1+\mathrm{d}y{e}_2$ is a tangent vector on $k_2$. The length L of $k_2$ with respect to $\tilde{g}$ is

$$\mathrm{d}L=\sqrt{\tilde{g}(\mathrm{d}v,\mathrm{d}v)}.$$

From (28) we find

$$\tilde{g}(\mathrm{d}v,\mathrm{d}v)=\sqrt{\frac{2cos\phi +1}{1+cos\phi }\mathrm{d}{x}^2+\frac{2cos\phi -1}{1-cos\phi }\mathrm{d}{y}^2}.$$

Then we obtain

$$L={\oint }_{k_2}\sqrt{\frac{2cos\phi +1}{1+cos\phi }\mathrm{d}{x}^2+\frac{2cos\phi -1}{1-cos\phi }\mathrm{d}{y}^2}.$$

(32)

We substitute

$$x=R\sqrt{\frac{1+cos\phi }{2cos\phi +1}}cost,y=R\sqrt{\frac{1-cos\phi }{2cos\phi -1}}sint,t\in [0,2\pi ]$$

into (32) and get

$$L={\int }_0^{2\pi }\sqrt{R^2{sin}^{2}t+R^2{cos}^{2}t}dt,$$

which implies (31). □

Now, we consider Case (B). The circle $k_2$ has a radius $R=ri,r>0,i^2=-1$ and $\phi$ satisfies

$$\frac{2\pi }{3}<\phi <\frac{3\pi }{4}.$$

(33)

By calculations similar to Case (A), but taking into account that R is an imaginary number, we find that the circle $k_2$ with (33) has an imaginary length (31).

4.1. Area of a circle with respect $\tilde{g}$

For Case (A) we state the following

Theorem 9. 

The area $\mathtt{A}$ of the circle $k_2$ with (30) and radius $R>0$ is

$$\mathtt{A}=\pi R^2.$$

(34)

Proof. 

Let us denote $\theta =\angle (e_1,e_2)$. The cosine of $\theta$ with respect to $\tilde{g}$ is

$$\widetilde{cos}\theta =\frac{\tilde{g}(e_1,e_2)}{\sqrt{\tilde{g}(e_1,e_1)}\sqrt{\tilde{g}(e_2,e_2)}}.$$

Then, using (28), we get $\widetilde{cos}\theta =0$, which implies

$$\widetilde{sin}\theta =1.$$

(35)

In the coordinate plane $p_{xy}$, we construct a parallelogram with locus vectors $\mathrm{d}x{e}_1$ and $\mathrm{d}y{e}_2$. For its area $\mathtt{A}$ with respect to $\tilde{g}$ we get

$$\mathrm{d}\mathtt{A}=\sqrt{\tilde{g}(\mathrm{d}x{e}_1,\mathrm{d}x{e}_1)}\sqrt{\tilde{g}(\mathrm{d}y{e}_2,\mathrm{d}y{e}_2)}\widetilde{sin}\theta .$$

We apply (28) and (35) in the above equality and get

$$\mathrm{d}\mathtt{A}=\frac{\sqrt{4{cos}^2\phi -1}}{sin\phi }\mathrm{d}x\mathrm{d}y.$$

(36)

We integrate (36) over the region $D_2$ inside $k_2$ and calculate

$$\mathtt{A}=\frac{\sqrt{4{cos}^2\phi -1}}{sin\phi }\int {\int }_{D_2}\mathrm{d}x\mathrm{d}y,$$

(37)

with

$$D_2:\frac{2cos\phi +1}{1+cos\phi }{x}^2+\frac{2cos\phi -1}{1-cos\phi }{y}^2\le R^2.$$

We substitute

$$x=\sqrt{\frac{1+cos\phi }{2cos\phi +1}}R\rho cost,y=\sqrt{\frac{1-cos\phi }{2cos\phi -1}}R\rho sint,t\in [0,2\pi ],\rho \in [0,1],$$

and Jacobian $▵={\displaystyle \frac{sin\phi }{\sqrt{4{cos}^2\phi -1}}}{R}^2\rho$ into (37). Finally we get (34). □

Now, we consider Case (B). The circle $k_2$ has an equation (29) with conditions (33) and a radius $R=ri,r>0,i^2=-1$. By calculations analogous to the previous case, we obtain that the area of $k_2$ is given in (34). This area $\mathtt{A}$ has a negative value.

4.2. Circulation and flux

We consider a closed curve $k_2$ in ${\beta }_2$, given by

$$x=x\left(t\right),y=y\left(t\right),t\in [\alpha ,\beta ],$$

(38)

where $x\left(\alpha \right)=x\left(\beta \right)$, $y\left(\alpha \right)=y\left(\beta \right)$.

Let

$$F(x,y)=P(x,y)e_1+Q(x,y)e_2$$

(39)

be a vector force field on the curve $k_2$.

We denote by $D_2$ the region inside $k_2$. For both cases (A) and (B) of the ellipse (29) the following statements are valid.

Theorem 10. 

The circulation C done by the force (39) along the curve (38) is expressed by

$$\begin{array}{c}\hfill C={\oint }_{k_2}\left(\frac{2cos\phi +1}{1+cos\phi }P(x,y)\mathrm{d}x+\frac{2cos\phi -1}{1-cos\phi }Q(x,y)\mathrm{d}y\right),\end{array}$$

(40)

where $\phi \in \left(\frac{\pi }{3},\frac{\pi }{4}\right)\cup \left(\frac{2\pi }{3},\frac{3\pi }{4}\right)$.

Proof. 

For the circulation C of a vector force field F acting along the curve (38) we use (21), where $v=x{e}_1+y{e}_2$ is the radius vector of a point on $k_2$. Therefore we have

$$C={\oint }_{k_2}\tilde{g}(F,\mathrm{d}v),$$

(41)

with a tangent vector $\mathrm{d}v=\mathrm{d}x{e}_1+\mathrm{d}y{e}_2$ on $k_2$. Then, by virtue of (28) and (39), we obtain

$$\begin{array}{c}\hfill \tilde{g}(F,\mathrm{d}v)=\left(\frac{2cos\phi +1}{1+cos\phi }P(x,y)\mathrm{d}x+\frac{2cos\phi -1}{1-cos\phi }Q(x,y)\mathrm{d}y\right).\end{array}$$

(42)

Hence (38), (41) and (42) imply (40). □

Theorem 11. 

The flux T of the vector field $\mathrm{curl}F$ across the region $D_2$ inside the curve (38) is expressed by

$$T=-2{cot}^3\phi \sqrt{4{cos}^2\phi -1}\int {\int }_{D_2}(Q_x-P_y)\mathrm{d}x\mathrm{d}y,$$

(43)

where $\phi \in \left(\frac{\pi }{3},\frac{\pi }{4}\right)\cup \left(\frac{2\pi }{3},\frac{3\pi }{4}\right)$.

Proof. 

We determine a vector w in $T_{p}M$ by the equality

$$w=\frac{1}{sin\phi \sqrt{1-2{cos}^2\phi }}\left({cos}^2\phi u-(cos\phi )Su+{sin}^2\phi S^{2}u\right),$$

(44)

where $\phi \in \left(\frac{\pi }{3},\frac{\pi }{4}\right)\cup \left(\frac{2\pi }{3},\frac{3\pi }{4}\right)$. Using (1), (2), (7) and (27) we verify that

$$g(w,e_1)=g(w,e_2)=0,g(w,w)=1.$$

The coordinate system $Oxyz$, such that $e_1\in Ox$, $e_2\in Oy$, $w\in Oz$ is orthonormal.

We obtain the curl of F, determined by (39), using the equality $\mathrm{curl}F=(Q_x-P_y)w.$ For the flux T of the vector field $curlF$ across the region $D_2$ inside the curve (38) we have

$$T=\int {\int }_{D_2}\tilde{g}(curlF,w)\mathrm{d}\mathtt{A}.$$

(45)

With the help of (28) and (44) we calculate

$$\tilde{g}(w,w)=-\frac{2{cos}^3\phi }{{sin}^2\phi }.$$

Then, from (36) and (45), it follows (43). □

Now, we introduce the following notations:

$$\begin{array}{c}\hfill c_1=\frac{2cos\phi +1}{1+cos\phi }{\oint }_{k_2}Pdx,c_2=\frac{2cos\phi -1}{1-cos\phi }{\oint }_{k_2}Qdy.\end{array}$$

(46)

On the other hand, due to Green’s formula, we have

$$\begin{array}{c}\hfill \int {\int }_D{P}_y\mathrm{d}x\mathrm{d}y=-{\oint }_{k}P\mathrm{d}x,\int {\int }_D{Q}_x\mathrm{d}x\mathrm{d}y={\oint }_{k}Q\mathrm{d}y.\end{array}$$

Bearing in mind the latter equalities we obtain the following statement.

Theorem 12. 

The relation between the circulation (40) and the flux (43) is determined by

$$\begin{array}{cc}\hfill T=-2{cot}^3\phi (& (1+cos\phi )\sqrt{\frac{2cos\phi -1}{2cos\phi +1}}{c}_1+(1-cos\phi )\sqrt{\frac{2cos\phi +1}{2cos\phi -1}}{c}_2),\hfill \end{array}$$

where $c_1$ and $c_2$ are given in (46).