# A Textbook Of Analytical Geometry Of Two Dimensions - download pdf or read online

By P. K. Jain, Ahmed Khalid

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Proposition. Let M be a non-degenerate hypersurface in An+1 . 1) M is an affine hypersphere. 2) B = L1 · G. 3) B = L1 · id. , n. Definition and Remark. Assume that x is locally strongly convex; that means that the Blaschke metric G is (positive) definite. In this case the affine Weingarten operator B has n real eigenvalues λ1 , λ2 , · · ·, λn , the affine principal curvatures. Then: (i) The relation B = L1 · G is equivalent to the equality of the affine principal curvatures: λ1 = λ 2 = · · · = λ n .

According to our notation in Riemannian geometry, κ(h) is the normed relative scalar curvature of the relative metric h. 4 Classical version of the fundamental theorem Uniqueness Theorem. Let (x, U, Y ) and (x , U , Y ) be non-degenerate hypersurfaces with the same parameter manifold: x, x : M → An+1 . Assume that h = h and A = A. Then (x, U, Y ) and (x , U , Y ) are equivalent modulo a general affine transformation. Existence Theorem. 5in Local Relative Hypersurfaces ws-book975x65 39 such that the integrability conditions in the classical version are satisfied.

Affine Gauß maps and Euclidean structure. In the case rank B = n it is often convenient to consider the two hypersurfaces, defined from the affine Gauß maps, as follows: We consider a Euclidean inner product , : V × V → R on V and identify V and V ∗ as usual. The three relations U, Y = 1, U, dY = 0, dU, Y = 0 imply that both affine Gauß indicatrices are a polar pair, that means they correspond via an inversion at the unit sphere. , en ]. , en ]. Using the Euclidean structure of V , we can express the conormal in terms of the Euclidean unit normal µ of x: 1 U = |K| n+2 · µ.