By Paul B. Yale

Designed for advanced undergraduate arithmetic or physics classes, this volume focuses on "practical geometry," emphasizing themes and methods of maximal use in all components of arithmetic. topics comprise algebraic and combinatoric preliminaries, isometries and similarities, an advent to crystallography, fields and vector areas, affine areas, and projective areas. 1968 version.

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**Example text**

Let x, x* be a polar pair and n = 3. Then the following statements are equivalent: (i) (M, I) is curvature homogeneous and H\ = constant; 42 (ii) (M, I*) is curvature homogeneous and H* = constant. 5. Corollary. Let x,x* : M -t S " + 1 ( l ) be polar hypersurfaces of dimension n > 3. Then the following statements are equivalent; (i) x is umbilical; (ii) x" is umbilical; (iii) (M, I) has constant sectional curvature; (iv) (M, f ) has constant sectional curvature. 6. Proposition. Let x, x" be polar.

1. Definition. Let (M, g) be a semi-Riemannian manifold and $ a Codazzi tensor. If the eigenvalues of $ are constant on M, $ is called an isoparametric Codazzi tensor. 2. Theorem. , en. ,n} befixed;then we have the Cartan identity J2 (Xi-\j)-lIC(ei,ej) =0 3 for g{ej,ek) = 6ik. 3. Corollary. Let x : M -> M(«) be a hypersurface in a space form M(K) of constant curvature K. ,fir. Then, for i € { 1 , . . 5. Weitzenboeck formulas. We recall two different versions of a Weitzenboeck for mula for traceless Codazzi tensors; see [WEG], [SI-1], [LIU-S-W]; [KOHL] contains other Weitzenboeck formulas.

3. Curvature relations b e t w e e n x and x'. ,n. For a polar pair x,x* : M -*• S " + 1 ( l ) we have: (i) The metric I is flat if and only if the polarized metric I* is flat; (ii) let 0 < r < n — 1 and r even; then H* is g-intrinsic (iii) let 0 < r < n - 1 and r odd; then {H*)2 is g-intrinsic and Hr is g*-intrinsic; and {Hr)2 is g'-intrinsic; (iv) if n=2 or 3