By Chuan-Chih. Hsiung

This helpful ebook comprises chosen papers of Prof Chuan-Chih Hsiung, well known mathematician in differential geometry and founder and editor-in-chief of a distinct foreign magazine during this box, the magazine of Differential Geometry.

During the interval of 1935-1943, Prof Hsiung was once in China engaged on projective differential geometry less than Prof Buchin Su. In 1946, he went to the U.S., the place he progressively shifted to international difficulties. Altogether Prof Hsiung has released approximately a hundred study papers, from which he has chosen sixty four (in chronological order) for this quantity.

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**Example text**

On the other hand, the four points of intersection, other than A and A*, of the asymptotic tangents of the surfaces S, S* at the points A, A* lie on two lines through the point P, namely, (10) x2 ± (lf^) "22^33/ x4 = 0, x3 = 0. 719 PROJECTIVE INVARIANTS From equations (9), (10) it follows at once that the cross-ratio of the four lines t, t*, (10), (9) is equal to (11) ±(±1)*(^) 1/4 ) = ±(±l)i71/4. (o, », J^f, \ \t22^33' \ ''11^22/ / Hence we obtain the following projectively geometrical characterization of the invariant I: Let S, S* be two surfaces in ordinary space having a common tangent plane at two ordinary points A, A*; t, t* the harmonic conjugate lines of AA* respectively with respect to the asymptotic tangents of the surfaces S, S* at the points A, A*; P the point of intersection of the tangents t, t*; and s any one of the two lines through the point P on which lie the four points of intersection, other than A and A*, of the asymptotic tangents of the surfaces S, S* at the points A, A*.

K2 and Kf may be respectively written as (7) and (40) The equations of af2Xi + atxxx3 — afx2x3 + k*x\ = 0, where k* is an arbitrary constant. From equations (7) and (40) it follows immediately that the equation of the four lines joining Ol to the four intersections of K2 and Kf is (41) a\x\ + 2a2x\x3 + • • • = 0 , the unwritten terms being of at least order 2 in x3 . It is easily seen that the polar line of any point on Oi0 2 with respect to the four lines denned by equation (41) is the line (8).

Let the polar spaces of the line 00* with respect to the asymptotic hypercones of the hypersurfaces F»_i, V*-\ a t the points 0, 0* be respectively denoted by /„_2, /n*-2, which determine a space tn-3 of n — 3 dimensions in the common tangent hyperplane x B + 1 = 0 . If the n — 2 vertices, other than 0 and 0*, of the system of reference in the hyperplane *n+i = 0 be chosen in the space tn-s, then the invariant / may be reduced to i n n / > » i l \ (n-2)/J (9) where Lnn, Mu are the minors of lnn, mu in the determinants L, M respectively.