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By Heiberg J.L. (ed.)

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Differential geometry and its applications: proceedings of the 10th International Conference, DGA 2007, Olomouc, Czech Republic, 27-31 August 2007

This quantity comprises invited lectures and chosen study papers within the fields of classical and smooth differential geometry, international research, and geometric tools in physics, awarded on the tenth foreign convention on Differential Geometry and its purposes (DGA2007), held in Olomouc, Czech Republic.

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Then, according to Ref. 632, we have:  2 j   ∂E X i = 2b2 f 2 gδε1 , ∂E ∂ ϕ = 2b2 f gδε1 ,  j  i ∂zε  ∂zη ∂uε ∂uη       ∂C i ∂ 2 E ∂ 2 ϕj   X i = 2b2 f (1 + 2g 2 ), X =0   ∂z i ∂zηj ∂uε ∂uη ∂zεi ε  ∂ 2 C 2 ∂ 2 ϕj   X i = 2f (−f h + 1 + g 2 ),   i ∂z j ∂uε ∂uη  ∂z  η ε      ∂C ∂ 2 ϕj g   (f h + 1 + g 2 )δε1  j ∂uε ∂uη = ∂zη 1 + g2 and hence from (21) we infer Hi X i = −1 3f 3 b2 (1 + 2g 2 ) h + 2g 2 (b2 f 2 − 2) + 3b2 f 2 − 2 , f 3 b2 1 + g2 whence the claim follows.

15 (1963) 121–139. 8. A. Sagle, Some homogeneous Einstein manifolds, Nagoya Math. J. 39 (1970) 81–106. 9. G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom. 8 (1973) 599–614. 10. A. Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987) 207–227. 11. M. Kerr, New examples of homogeneous Einstein metrics, Michigan J. Math. 45 (1998) 115–134. 12. V. Alekseevsky, I. Dotti and C. Ferraris, Homogeneous Ricci positive 5manifolds, Pacific J. Math. 175 (1996) 1–12.

Det(H) · (1 + ls ls ). b) We have det(H) We note that (4) briefly rewrites ˜ + 2νbv + ρ ≤ 0 Q : v t Hv ˜ where v = (v 1 , . . , v n )t , H ⇔ ˆ (v t , 1)H = C t C, C v 1 ≤0 (5) (zαi )i,α=1,n , b = √ µb = (z1n+1 , . . , znn+1 ) and ˆ = H ˜ νbt H νb ρ n+1 ˜ ab ) (h )α a,b=1,n ν(zα = ν(zαn+1 )αt ρ . (6) After performing an orthogonal transformation, the bounding quadric ∂Q has a canonic analytic equation; hence n n λa (v a )2 + Q: a=1 ∆ ≤0⇔ δ∗ a=1 ( (v a )2 ≤ 1, (v 1 , . . , v n ) ∈ Rn , −∆/(δ∗ λa ))2 ˜ δ∗ = λ1 · · · · · λn = det H ˜ and where λ1 , .

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