By J. A. Thorpe

Some time past decade there was an important switch within the freshman/ sophomore arithmetic curriculum as taught at many, if no longer such a lot, of our faculties. This has been caused by way of the advent of linear algebra into the curriculum on the sophomore point. some great benefits of utilizing linear algebra either within the educating of differential equations and within the educating of multivariate calculus are via now widely known. a number of textbooks adopting this perspective are actually to be had and feature been greatly followed. scholars finishing the sophomore yr now have a good initial less than status of areas of many dimensions. it's going to be obvious that classes at the junior point may still draw upon and toughen the techniques and talents discovered throughout the prior yr. regrettably, in differential geometry at the very least, this is often often no longer the case. Textbooks directed to scholars at this point often limit consciousness to 2-dimensional surfaces in 3-space instead of to surfaces of arbitrary size. even though many of the contemporary books do use linear algebra, it is just the algebra of ~3. The student's initial realizing of upper dimensions isn't cultivated

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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.