The form of the spheres of some left invariant nonholonomic inner metrics on SL₂R and SO(3)
by
Irina Zubareva
Alexandrovna

We define the distance d_K on the bundle of unit vectors V_K over Lobachevsky (Riemann) plane M_K with curvature K; d_K(v₁, v₂) is equal to the greatest lower bound of the lengths for curves, whose horizontal lifts relatively to Levi-Civita connection join v₁ and v₂. So we get left invariant nonholonomic inner metrics on Lie groups SL₂(R) (SO(3)). We find the exact forms of spheres in (V_K, d_K) by means of known solutions for isoperimetric problem on M_K. The author use coordinate system (r, \alpha, \beta), 0 <= r, -\pi <= \alpha < \pi, -\pi <= \beta < \pi, where r is the distance from the origin of fixed vector v_O in V_K to the origin of given vector v in V_K; \alpha, \beta are some angle coordinates.

Theorem 1.  The diameter of the space (V_K, d_K) is equal to \surd3\pi\sigma, where K=\sigma^-2, and every sphere of radius \surd3\pi\sigma is the point. The sphere of radius T < \surd3\pi\sigma is a revolution surface of that part of the curve determined by parametric equations

r= +/- 2\sigmaarcsin(\frac1tsin\fracTt2\sigma),    \alpha = +/- \alpha(t),    1 <= t <= \frac2\pi\sigmaT,

which lays in the zone -\pi <= \alpha < \pi on the plane \beta = 0, around the axis \alpha, and

1. If T < \pi\sigma, then

\alpha(t)= ì ï í ï î 2 ê ê \fracT2\sigma Ö   t2-1   -arcsin\frac Ö   t2-1   sin\fracTt2\sigma Ö   t2-sin2\fracTt2\sigma   ê ê , if 1 <= t < \frac\pi\sigmaT, 2 æ è \pi-\fracT2\sigma Ö   t2-1   -arcsin\frac Ö   t2-1   sin\fracTt2\sigma Ö   t2-sin2\fracTt2\sigma   ö ø , if \frac\pi\sigmaT <= t <= \frac2\pi\sigmaT.

2. If \pi\sigma <= T < \surd3\pi\sigma, then

\alpha(t)=2 æ è \pi-\fracT2\sigma Ö   t2-1   -arcsin\frac Ö   t2-1   sin\fracTt2\sigma Ö   t2-sin2\fracTt2\sigma   ö ø .

Theorem 2.  The sphere of the space (V_K, d_K), K=-\sigma^-2, of any radius T is a revolution surface of that part of the closed curve determined by aggregate of parametric equations r= +/- r(t), \alpha = +/- \alpha(t), 0 <= t <= \frac2\pi\sigmaT, and r= +/- r(u), \alpha = +/- \alpha(u), 0 < u <= 1, which lays in the zone -\pi <= \alpha <= \pi on the plane \beta = 0, around the axis \alpha, where

r(t)=2\sigmaln æ è \frac1tsin\fracTt2\sigma+\frac1t Ö   t2+sin2\fracTt2\sigma   ö ø ,    0 <= t < \frac2\pi\sigmaT,

\alpha(t)= ì ï í ï î 2 æ è \fracT2\sigma Ö   1+t2   -arcsin\frac Ö   t2+1   sin\fracTt2\sigma Ö   t2+sin2\fracTt2\sigma   ö ø , if 0 <= t <= \frac\pi\sigmaT, 2 æ è \fracT2\sigma Ö   1+t2   -\pi+arcsin\frac Ö   t2+1   sin\fracTt2\sigma Ö   t2+sin2\fracTt2\sigma   ö ø , if \frac\pi\sigmaT < t <= \frac2\pi\sigmaT.

r(u)=2\sigmaln æ è \frac1ush\fracTu2\sigma+\frac1u Ö   u2+sh2\fracTu2\sigma   ö ø ,    \alpha(u)=2 æ è \fracT2\sigma Ö   1-u2   -arcsin\frac Ö   1-u2   sh\fracTu2\sigma Ö   u2+sh2\fracTu2\sigma   ö ø .

The author was supported by the INTAS (Grant 97-10170).

Date received: January 26, 2000