By Thomas E. Cecil

ISBN-10: 0387977473

ISBN-13: 9780387977478

ISBN-10: 1475740964

ISBN-13: 9781475740967

**Lie Sphere Geometry** presents a contemporary remedy of Lie's geometry of spheres, its fresh functions and the examine of Euclidean area. This ebook starts off with Lie's building of the distance of spheres, together with the elemental notions of orientated touch, parabolic pencils of spheres and Lie sphere transformation. The hyperlink with Euclidean submanifold thought is proven through the Legendre map. this gives a robust framework for the learn of submanifolds, in particular these characterised through regulations on their curvature spheres. Of specific curiosity are isoparametric, Dupin and taut submanifolds. those have lately been categorised as much as Lie sphere transformation in definite distinct circumstances throughout the advent of normal Lie invariants. the writer offers whole proofs of those classifications and exhibits instructions for extra learn and wider software of those tools.

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**Example text**

L. L intersects Qn+-l in a Moebius space. We now show that any line on the quadric intersects such a Moebius space in exactly one point. 5: Let [z] be a timelike point in IPn+2 and f a line which lies on n+1 Q . L in exactly one point. Proof: Any line in projective space intersects a hyperplane in at least one point. L. L cannot contain the 2-dimensionallightlike vector space which projects to f . 0 As a consequence, we obtain the following corollary. 6: Every parabolic pencil contains exactly one point sphere.

0 is equivalent to the condition p. , ~ is Hence the parabolic pencil of spheres corresponding to e The fact that < kh k2 > n tangent to S at p. can be identified with the point ( p, ~ ) in T1Sn. The points on the line be parametrized as [ K t] = [cos t kl + sin t kz) = [(cos t, cos t P + sin t ~ , sin t)] . 4) Pt = cos t P + sin t ~ , ecan 28 and signed radius t. 1. Lie Sphere Geometry These are precisely the spheres through p in oriented contact with the great sphere corresponding to [kiJ. Their centers lie along the geodesic in Sn with initial point p and initial velocity vector 1; .

E3, ... ,en+2}. 4 Hyperspheres in Sn and Hn n+l { Y E 1R1 I (y, y) = -1 , 21 Yl ~ 1 } , on which the restriction of the Lorentz metric ( , ) is a positive definite metric of constant sectional curvature -1 (see Kobayashi-Nomizu [1, Vol. II, pp. 268-271] for more detail). The distance between two points p and q in given by d( p, q ) n = cosh-1(- ». 6) =- cosh P . ( p, Y ) . n n+l . As before With S , we first embed 1R1 Let p E n lOto IP n n+l by the map 'I'(y) H and let z =y + e2 for Y E H . Then we have ( p, Y ) = ( z, p ) / ( z, e2 ) .

### Lie Sphere Geometry: With Applications to Submanifolds by Thomas E. Cecil

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