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

By John Casey

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Extra resources for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

Example text

To continue, antisymmetrise 0= c2 d2 a2 =2 c2 d2 a2 b2 b1 d 1 g¯ij S ia2 b1 b2 S jc g¯ij S ia2 b1 b2 S jc 2 d1 d2 2 d1 d2 + S ia2 b2 d1 S jc 2 b1 d2 + S ia2 d1 b1 S jc 2 b2 d2 in a2 , b2 , c2 , d2 . Then the last term vanishes by the symmetry of S j c2 b2 d2 in b2 , d2 and yields 0= a2 b2 c2 d2 g¯ij S ia2 b1 b2 S jc 2 d1 d2 + S ia2 d1 b2 S jc 2 b1 d2 . Both sum terms are equal under antisymmetrisation in a2 , b2 , c2 , d2 and contraction with g¯ij . Indeed, exchanging b1 and d1 is tantamount to exchanging a2 with c2 and b2 with d2 and renaming i, j as j, i.

3 Isokernel planes and integrable Killing tensors from S2 K.

1 Decomposition . . . . . . . . . . . 2 The action of the isometry group . . . . . . 3 Aligned algebraic curvature tensors . . . . . . 4 Diagonal algebraic curvature tensors . . . . . 5 The residual action of the isometry group . . . . 2 Solution of the algebraic integrability conditions . 1 Reformulation of the first integrability condition . . 2 Integrability implies diagonalisability . . . . . 3 Solution of the second integrability condition . . . 4 Interpretation of the Killing-St¨ ackel variety .

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