By Luther Pfahler Eisenhart
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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. To find the regularity conditions of the submanifolds. : x,-x2 YI-Y2 =1-Z2 X2-XI Y2-Y1 Z2-ZI Under the assumption that grad 't # 0 (i = 1, 2), the matrix A has rank less than 3 if and only if the third row is a linear combination of the first and second rows.
As 82 = 0, we can write that B = µ(e3 + e4). s - e4 sin a2s), where 4(A2 + µ'-) = 1. Integrating i 1, we get r(s) = r(0) + 2A a, (e1 sin al s +e2 cos a, s) + 2µ (e3 sin a2s + e4 cos a2 s). a2 In case (b) we have the equations: A2 = 0, (AB) = 0, 2(AB) + B2 = 0. The derivative of , has the form i ds = i[(Ae''I- Ae-(" s)a + (BeI'23 - Be-'"=s)a,l. 2 1, then () =const. By Frenet formulas, 41/ds=k16. As k1 =const and Therefore at2 A2 e2in's+c Ate-2n's+a2B2e2;"zs+a2B2a-3;023-ai2(AA) + (AB)e'("'+n')3 - 2(A1&)a1a2e'(n'-n2)s + 2(AB)e-("'+"=)s - 2(BB)a4 = const, - 2(AB)e-'(0'+0')5 CURVES 15 which produces the auxiliary condition - the term with exponent 2ia2s = i(a1 - a2)s is equal to zero.
Hence, the linear span of grad &, coincides with N. Let us consider some examples. Example 1 Hinge deformation manifold The hinge consists of constant length segments freely rotating around an axis. Suppose the segments of the hinge move in E3 in such a way that the endpoints of the segments move along the fixed surfaces. The particular case when all endpoints move along the fixed sphere has been proposed by Pogorelov (see Figure 5). To simplify the situation, consider the motion of one segment with its endpoints in a fixed surface.