Abstracts
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.
Flat torus; constant mean curvature; circular tori; stereographic projection
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.
Toro plano; Curvatura média constante; Toro circular; Projeção estereográfica
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
Correspondence Correspondence to E-mail: abbarros@mat.ufc.br AMS Classification: Primary 53A05, 53A10; Secondary 53A30.
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} ®![](https://minio.scielo.br/documentstore/1678-2690/G3syVWFX9H97Ht55tzBn9fw/c8dac39f38fa68233ffbfacf85f1fc7c1a8c10f4.gif)
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THEOREM 1. Let T2Ì 3 be a circular torus of constant mean curvature. Then
T2 = r1(T(sec a,tan a)) = S1(cos a) × S1(sin a).
Moreover, the mean curvature of T 2 is given by
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2 PRELIMINARIES
For an immersion f : M ® between Riemannian manifolds we will denote by d
the induced metric on M by f. Now let Mn,
and
be Riemannian manifolds, where the superscript denote the dimension of the manifold. Consider y : Mn ®
be an immersion, r :
®
a conformal mapping and set j = r ºy. Let f : M ®
be a function verifying d
= e2fd
. If
i and ki denote the principal curvatures of y and j = r °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.
LEMMA 1. Let y = (y1, y2, y3, y4) : M2® 3 \ {n} be an immersion of a surface M2, set j = r ºy and suppose d
where g = án, jñ denotes the support function on M2Ì 3.
PROOF. If we put y = y(u1, u2) then a direct computation gives
where l = (1 y4)1 = . So we can write d
= e2fd
with ef =
. Thus if n denotes a unit normal vector field to j(M2) then n = efx, where x stands for a unit normal vector field to y(M2). Hence we have from (1)
as we wished to prove.
3 PROOF OF THE THEOREM
PROOF. First we note that the circle G = {(x, 0, z) Î 3 : (x r)2 + z2 = a2} can be parametrized by the map g : [0, 2p] ®
3 defined by
In fact, it is enough to check that
Representing by Rq a rotation on 3 around the z axis, we see that Rq(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 = ru1/s2 and t = ru2/as. We note that such a choice implies 0 < u1< (2ps2)/r and 0 < u2< (2pas) / r. Let us call Rq(g(t)) of j(u1, u2), i.e.
Hence we have
where q(t) = a(s2 1) sin t + r(s2 + 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 k1 = and k2 =
. Taking into account (5), (7) and (8) we conclude from Lemma 1 that
Now we have that is constant if and only if s2 = 1. Moreover, s2 = 1 yields
=
(a2 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 òT(r, a) H2dA 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 òT(r, a)H2dA 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 òT(r, a)H2dA = òT(sec a, tan a)
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 r1T(a, r) is flat if and only if it has constant mean curvature. We notice if we set y = r1j where j was given by (4) then we have
where q(t) = a(s2 1) sin t + r(s2 + 1), (see(5)). Hence by using (3), (5), (6) and putting z = u1 + iu2 we conclude that
According to our theorem the metric d is flat if and only if r1T(r, a) has constant mean curvature in
3. In this case we have
i.e. r1T(r, a) is isometric to the product of circles S1 () × S1 (
). We note that this yields cos a =
and sin a =
, i.e. r(S1 (cos a) × S1 (sin a)) = T(sec a, tan a).
ACKNOWLEDGMENTS
This work was partially supported by FINEP-Brazil.
Manuscript received on May 30, 2003; accepted for publication on June 14, 2004; presented by MANFREDO DO CARMO* * Member Academia Brasileira de Ciências
- ABE N. 1982. On generalized total curvature and conformal mappings. Hiroshima Math J 12: 203-207.
- 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.
- NOMIZU K AND SMYTH B. 1969. A formula of Simons' type and hypersurfaces with constant mean curvature. J Diff Geom 3: 367-377.
- WILLMORE T. 1982. Total curvature in Riemannian geometry, Ellis Horwood limited, 168 pp.
Publication Dates
-
Publication in this collection
22 Nov 2004 -
Date of issue
Dec 2004
History
-
Accepted
14 June 2004 -
Received
30 May 2003