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Xm_ 1 ) = xm_ 1 • The level hypersurfaces are "horospheres" - that is, they are isometric to ( m-2)- dimensional Euclidean spaces. There is no focal variety. 0 Example 2. 3. 6. In Example 2. 1. 4 there is one focal variety which is isometric to R m-p. 1. 5 there are two focal varieties, one is isometric to sP-1 and the other is isometric to sQ-1. 6 there is one focal variety isometric to HP-1. 1) where u, v E R, X, Y, z E F, F is one of R, 0:, quaternions or Cayley numbers, and is conjugation on F.

3. 5) is zero. 4). We study M , the double cover of M , which locally can be described as the set s of points eiO (coss x + isins y), s S E (0, n/4), (x,y) E Sn+l, 2 and 0 E (0,2 n). Fix 0, then Sn+l 2 is embedded in a;n+1 as the manifold S s . Fix coss x + • n+l, 2 isins y, then as 0 varies, we trace out a great circle of s2n+1. How does this eire le intersect the S s ? 6) bothofwhicharecontainedin Ms, where x(O) = x 0 ,y(O) = y 0 , 9(0) = 9 0 and (x(u), y(u)) E Sn+l, 2 for all u. These are both curves through p, and without loss of generality 6 y 1 (0) 1( = i~ = e 0) = 1.

Cartan classified all such families of isoparametric hypersurfaces. He also solved the classification problem for p = 3 on S m-1 [ 6 I, finding that m could be only 2, 5, 8, 14 or 26. M~zner [ 30 I showed that every isoparametric hypersurface is algebraic, and only certain p are allowed; we state his remarkable theorem without proof. 5 [301 Let M be sm- 1 and Me an isoparametric hypersurface with p distinct principal curvatures ~1 , ••• , ~p with multiplicities m 1 , m respectively. e. there are at most 2 distinct multiplicities.

### Answers Sydsaeter & Hammond - Mathematics for Economic Analysis

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