A treatise on the analytical geometry of the point, line, by John Casey

By John Casey

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Tromba [2 ] . Now i t is c l e a r t h a t a smooth Jordan curve r C ~ K(F) ~ 47 has to be unknotted, 3 with total but a curve r w i t h t o t a l curvature K(F) = 4~+s 0 , s o > 0 , can be a t o r u s knot o f a r b i t r a r y (2,2k+i), curvature type k c ~. So one can be i n t e r e s t e d whether the p o s s i b i l i t y t h a t F bounds more then N minimal surfaces o f the type o f the d i s c is r e l a t e d to the class to which F belongs. The answer is negative and we s t a t e i t knot in the f o l l o w i n g way: Theorem 3: For any N ~ , any k ~ ~, and any s > 0 t h e r e e x i s t s a r e g u l a r real a n a l y t i c Jordan curve F in ~ 3 , F being a torus knot of type (2,2k+1), such t h a t F has t o t a l curvature less then 4~+s and bounds more than N minimal surfaces o f the type o f the d i s c .

6 C~(M) , we get the same result. 4. If H ~ const. eigenvalues spaces. 1 , then Lemma. tool intervall following for this Assume H 6 C2(M). 2) gives 0 = 2 ] E X i j ( U i - O j ) 2 do - # llfijk i<3 - ~ {~ij f i j f + 2 u i f i j f j 2 do - + P ][fij [12} d o . 3) below. n >_ 3 , ~. 2 identity >ijfijf : [3], p. 3 do Aijk of Obata ~ ~ const. [4], Tanno , we define ~ £ C2(M), the [8] and B e n k o et al. for an e i g e n f u n c t i o n [1] f with (3,0) tensor :: fijk + (a+b~)gijfk + (c+d~) (gikfj + gjkfi ) ' a, b, c, d are real functions.

Now if g is the G-invariant metric on G/K corresponding to the scalar product on P @ Q given by the two preceding scalar products on P and Q and the condition that P and Q are orthogonal, then~rg is a Riemannian submersion from (G/K,g) onto (G/H,h) with totally geodesic fibres isometric to (H/K,k). This construction gives many examples, including the preceding one. General construction: let ~ : M ~ B be a locally trivial fibration with typical fibre F and group G, supposed to be a Lie group of transformations of F, leaving some 31 Riemannian metric invariant.

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