**MATHEMATICAL SCIENCES**

**A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere **

**Abdênago Barros**

Departamento de Matemática-UFC, Bl 914, Campus do Pici, 60455-760 Fortaleza, CE, Brasil

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**ABSTRACT**

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

**Key words:** Flat torus, constant mean curvature, circular tori, stereographic projection.

**RESUMO**

Neste artigo mostraremos que a imagem inversa pela projeção estereográfica de um toro circular de revolução no espaço euclidiano de dimensão 3 tem curvatura média constante se e somente se os seus raios são o cateto e a hipotenusa de um triângulo retângulo apropriado.

**Palavras-chave:** Toro plano, Curvatura média constante, Toro circular, Projeção estereográfica.

**1 INTRODUCTION**

We will denote by *T*(*r, a*) the standard circular torus of revolution in ^{3} obtained from the circle G in the *xz* – *plane* centered at (*r*, 0, 0) with radius *a* < *r*, i.e.

Now let r : ^{3} \ {*n*} ® ^{3} be the stereographic projection of the Euclidean sphere ^{3} = {*x* Î ^{4}: *| x *|^{2} = 1}, where *n* = (0, 0, 0, 1) is its north pole. The inverse image of a circular torus in ^{3} under the stereographic projection will be called a circular torus in ^{3}. We would like to know when circular tori in ^{3} comes from constant mean curvature circular tori in ^{3} under the stereographic projection. A circular torus in ^{3} meant that it is obtained from a revolution of a circle in ^{3} under a rigid motion. A general *T*(*r, a*) will not satisfy the above requirement. For instance, it was proved by Montiel and Ros (Montiel and Ros 1981) that a compact embedded surface *S* with constant mean curvature contained in an open hemisphere of ^{3} must be a round sphere. Hence for *T*(*r, a*) contained inside or outside of the unit ball *B*(1) Ì ^{3}, r^{–1}(*T*(*r, a*)) will be contained in an open hemisphere of ^{3} and can not have constant mean curvature. Then among all tori *T*(*r, a*) which intercept the inside and the outside of the unit ball *B*(1) we will describe those which have the desired property. We will show that to construct such a torus we take an arbitrary point *P*(a) = (cos a, 0, sin a) on the unit circle of the *xz – plane*, 0 < a < p/2, draw its tangent until it meets the *x* axis at the point (a) = (sec a, 0, 0) which will be the center of the circle G whereas its radius will be *a* = tan a, i.e. the torus *T*(sec a, tan a) will satisfy the previous requirement. We note if *O* denotes the origin of ^{3} then the triangle *O P * is a right triangle. This description will yield that the Clifford torus is associated to a right triangle with two equal sides. More precisely, our aim in this note is to present a proof of the following fact:

THEOREM 1. *Let T*^{2} Ì ^{3} be a circular torus of constant mean curvature. Then

*T*^{2} =* *r^{–1}(*T*(sec a,tan a)) = *S*^{1}(cos a) × *S*^{1}(sin a).

*Moreover, the mean curvature of T*^{2}* is given by ** =* *. *

**2 PRELIMINARIES**

For an immersion *f* : *M* ® between Riemannian manifolds we will denote by *d* the induced metric on *M* by *f*. Now let *M ^{n}*, and be Riemannian manifolds, where the superscript denote the dimension of the manifold. Consider y :

*M*® be an immersion, r : ® a conformal mapping and set j = r

^{n }_{º}y. Let f :

*M*® be a function verifying

*d*=

*e*

^{2f}

*d*. If

_{i}and

*k*denote the principal curvatures of y and j = r

_{i}_{°}y, respectively, then we get

where x is a unit normal vector field to y(*M*), see for instance (Abe 1982) or (Willmore 1982). At first we will recall the following known lemma of which we sketch the proof.

*Let*y = (y

_{1}, y

_{2}, y

_{3}, y

_{4})

*: M*

^{2}®

^{3}\ {

*n*}

*be an immersion of a surface M*

^{2},

*set*j = r

_{º}y

*and suppose d*

*= e*

^{2f}

*d*

*. Then we get*

*where g = *á*n, *jñ* denotes the support function on M*^{2} Ì ^{3}.

PROOF. If we put y = y(*u*_{1}, *u*_{2}) then a direct computation gives

where l = (1 – y_{4})^{–1} = . So we can write *d* = *e*^{2f}*d* with e^{f} = . Thus if n denotes a unit normal vector field to j(*M*^{2}) then n = *e*^{–f}x, where x stands for a unit normal vector field to y(*M*^{2}). Hence we have from (1)

as we wished to prove.

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**3 PROOF OF THE THEOREM**

PROOF. First we note that the circle G = {(*x*, 0, *z*) Î ^{3} : (*x – r*)^{2 }+ *z*^{2} = *a*^{2}} can be parametrized by the map g : [0, 2p] ® ^{3} defined by

In fact, it is enough to check that

Representing by *R*_{q} a rotation on ^{3} around the *z – axis*, we see that *R*_{q}(g(*t*)) is a circular torus *T*(*r, a*) if g is a parametrization of the circle G given above. We put now s = , q = *ru*_{1}/s^{2} and *t* = *ru*_{2}/*a*s. We note that such a choice implies 0 __<__ *u*_{1} __<__ (2ps^{2})/*r* and 0 __<__ *u*_{2} __<__ (2p*a*s) / *r*. Let us call *R*_{q}(g(*t*)) of j(*u*_{1}, *u*_{2}), i.e.

Hence we have

where *q*(*t*) = *a*(s^{2 }– 1) sin *t* + *r*(s^{2 }+ 1). Now a straightforward computation yields

From that we derive that j is a conformal parametrization of *T*(*r, a*) satisfying

Moreover, a unit vector field normal to j is given as follows:

Therefore we conclude that

On the other hand a new computation gives us

From this we have *k*_{1} = and *k*_{2} = – . Taking into account (5), (7) and (8) we conclude from Lemma 1 that

Now we have that is constant if and only if s^{2} = 1. Moreover, s^{2} = 1 yields = (*a*^{2 }– 1). Since *a* < *r* we put *a* = *r* sin a, *r* = sec a and this completes the proof of the theorem.

We point out that = 0 if and only if *a* = 1 and *r* = which corresponds to the right triangle with two equal sides.

**4 THE WILLMORE MEASURE ON T(r, a) **

In this section we will present a simple way to compute ò_{T}_{(r, a) }*H*^{2}*dA* by using the parametrization of *T*(*a, r*) given by (4). We observe that if *dA* denotes the element of area of *T*(*r, a*) then its Willmore measure is given by

Hence, using Gauss-Bonnet theorem, we easily conclude that

]]> Therefore the family of tori*T*(

*a*,

*a*) , which corresponds to the family of right triangles with two equal sides, yields the minimum for ò

_{T}_{(r, a)}

*H*

^{2}

*dA*among all circular tori. Moreover, from (9) its value is (see also Willmore 1982)

Since *a* < *r*, if we choose a such that sin a = , we conclude from (9) the following corollary.

COROLLARY 1. *Given a circular torus T*(*r, a*) Ì ^{3}* we have a circular torus T *(sec a, tan a) Ì ^{3}* such that ò _{T}*

_{(}

_{r, a}_{)}

*H*

^{2}

*dA =*ò

_{T}_{(sec a, tan a)}

*dA*

_{a}

*. In other words, the family of circular tori with constant mean curvature in*

^{3}

*cover all values of*ò

_{T}_{(}

_{r, a}_{)}

*H*

^{2}

*dA.*

**5 CONCLUDING REMARKS**

We point out that Theorem 2 of K. Nomizu and B. Smyth (Nomizu and Smyth 1969) guarantees that a flat torus of constant mean curvature in ^{3} is isometric to a product of circles. Then r^{–1}*T*(*a, r*) is flat if and only if it has constant mean curvature. We notice if we set y = r^{–1}j where j was given by (4) then we have

where *q*(*t*) = *a*(s^{2} – 1) sin *t* + *r*(s^{2 }+ 1), (see(5)). Hence by using (3), (5), (6) and putting *z* = *u*_{1 }+ *iu*_{2} we conclude that

*d*is flat if and only if r

^{–1}

*T*(

*r, a*) has constant mean curvature in

^{3}. In this case we have

i.e. r^{–1}*T*(*r, a*) is isometric to the product of circles *S*^{1} () × *S*^{1} (). We note that this yields cos a = and sin a = , i.e. r(*S*^{1 }(cos a) × *S*^{1 }(sin a)) = *T*(sec a, tan a).

**ACKNOWLEDGMENTS**

This work was partially supported by FINEP-Brazil.

**REFERENCES**

ABE N. 1982. On generalized total curvature and conformal mappings. Hiroshima Math J 12: 203-207. [ Links ]

MONTIEL S AND ROS A. 1981. Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures, A symposium in honor of Manfredo do Carmo, Edited by B. LAWSON AND K. TENEBLAT, Pitman Monographs 52: 279-296. [ Links ]

NOMIZU K AND SMYTH B. 1969. A formula of Simons' type and hypersurfaces with constant mean curvature. J Diff Geom 3: 367-377. [ Links ]

WILLMORE T. 1982. Total curvature in Riemannian geometry, Ellis Horwood limited, 168 pp. [ Links ]

** Correspondence to**

E-mail: abbarros@mat.ufc.br

AMS Classification: Primary 53A05, 53A10; Secondary 53A30.

Manuscript received on May 30, 2003; accepted for publication on June 14, 2004; presented by MANFREDO DO CARMO*

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