By F. W. J. Olver
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Extra resources for Asymptotic and special functions
_0 :(M,g)- (N,h), then s 0 = 0 if and only if dim M = 2 and 0 is conformal. 3 (N,h) is a harmonic map of the 2-sphere into a Riemannian manifold N, then 0 is conformal. Proof Choose local isothermal coordinates z on s 2 , so that g = p(z) dz dz. Then a tensor T E ~(0 2 T* M) has a type decomposition of the form T = T ( 2 ' 0 ) + T ( 1 • 1 ) + T(0, 2 ), where T(i,j) is spanned by dzi dzj locally. A calculation shows that (0*h)(1, 1 ) =e(0)g. Thus ~ 0 =s/ 2 •0) + s~0,2), where sJ 0 • 2) = s/2,0).
Furthermore, since the integral submanifolds of the S. m J distributions are totally geodesic in the level hypersurfaces of f, we can suppose Vx. Xi is proportional to ~ for i = 1, ••• , m. 1 Now n = 'Y 9 for some j = 1, • . , m, so the horizontal space through x is j • i1 j S. (x), and the vertical space 1-' 1 = X :Jt'X = S. (x) e R ~. 4) we see J that TI is horizontally conformal if and only if the number of distinct principal curvatures p is equal to 2. To see that n is harmonic, we work out trace V d n.
7 Tbe level hypersurfaces M , locally can be described as an "angu- lar product" of s1 with S~+l, 2 s , where the angle of incidence is 2 s - 11/2, and the plane of incidence is spanned by the tangent vectors to the curve p. ( u) of ( 2. 3. 4) and the curve y ( u) of ( 2. 3. 6). Example 2. 3. P. P. = 2 6 .. •• , n. x,x> 2 ,xER 21 , i 1 defines an isoparametric function f = F 1s21-1 on s 21 - 1 of degree 4, with the multiplicities of the principal curvatures being (m 1 ,m2 ). 5 is a special case of this example.