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The notes from a suite of lectures writer brought at nationwide Tsing-Hua collage in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of booklet "Thing Hua Lectures on Geometry and Analisys".
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Additional info for Mathematical foundations of quantum statistics. Translation from the 1st (1951) Russian ed
X") the coefficients of the second fundamental form are L; = z,,,,/ l + (grad z12. 3) 2 - (x4) 2 the asymptotic lines satisfy the equation (dx') 2 + (dr 2) 22 - (d%3) 2 - (dr4) 2 = 0. The p-parametric curve x1 =a cos a, gyp, x2 = a sin a, cp, a; = b COs a2cp, x4 = b sin a2rp, with a. b, a,, a2 constants, satisfy the equation above when a condition (aa,) 2 = (bat) 2 is satisfied. These curves are closed when a, /a2 is a rational number. The curve considered in the example lies in a sphere of radius v/a2 -+b2.
Let xi, y,, zi (i = 1, 2) be Cartesian coordinates of the endpoints. The location of the segment can be described by six coordinates x1, y1, 21,x2, }'2, z2. They fix the point in E6. Subject this coordinates to conditions: ''r(xny,, 21) = 0, i=1,2; 4'3 = 2 [(xi - x2)2 + U'1 -1'2)2 + (21 - 22)2 - a2]= 0. GENERAL PROPERTIES OF SUBMANIFOLDS 25 IIGURE 5 The first and second conditions are the equations of the surfaces to which the endpoints of the segments belong. So, we have an implicit representation of some submanifold in E6.
By means of the Weingarten decomposition, do = n,,, du' _ -L; dit'gikr,rk. For the principal direction the following holds: (L;,,-Agy)du'=0. Hence do = -Agydug/kr". = -Abler,,. du' = -A dr. Thus, Rodrigues' formula holds -- the differential of a unit normal along the principal direction is parallel to this direction: do = -A dr. l)2)4' where (') means differentiating with respect to t. Prove that coordinate curves are lines of curvature. 4 Integrability Condition for Principal Directions In contrast to the two-dimensional case, many-dimensional hypersurfaces do not, in general, possess coordinates that the coordinate hypersurface would be orthogonal to the principal directions.