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By Lesniewicz Paul

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Theorem 1. Let S be an n-surface in IRn + I, let ex: I -+ S be a parametrized curve in S, let to E I, and let v E S«(to)' Then there exists a unique vector field V, tangent to S along ex, which is parallel ~nd has V(t o) = v. PROOF. We require a vector field V tangent to S along ex satisfying V' = O. But V' = V"- (V . N ex)N ex = V - [(V • N ex)' - V • N ~ ex]N ex = V + (V • N ~ ex)N ex 0 0 0 0 0 so V' = 0 if and only if V satisfies the differential equation (P) This is a first order differential.

Let Hp = {T E Gp: T= P(I. for some piecewise smooth ct: [a, b] -+ 8 with ct(a) = ct(b) = pl. ' and (ii) for each ct in 8 from p to p there is a p in 8 from p to p such that P p = P; 1. 8. Let ct: I -+ 8 be a unit speed curve in an n-surface 8, and let X be a smooth vector field, tangent to 8 along ct, which is everywhere orthogonal to ct (X(t)) . &(t) = 0 for all t E 1). Define the Fermi derivative X' of X by X'(t) = X'(t) - [X'(t) • &(t)]&(t). 'X + fX' for all smooth functions f along ct, and (iii) (X .

Rn + l , a =1= O. Show that the spherical image of an nsurface S is contained in the n-plane al Xl + ... + an + 1 X n + 1 = 0 if and only if for every PES there is an open interval I about 0 such that p + ta E S for all tEl. 8. Show that if the spherical image of a connected n-surface S is a single point then S is part or all of an n-plane. [Hint: First show, by applying the corollary to Theorem 1, Chapter 5, to the constant vector fields W(q) = (q, w), where w 1. Rn + l: x • v = p • v}. 9. Rn + 1 -+ R is a smooth function such that Vf(p) =1= 0 for all pES.

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