By CK-12 Foundation

CK-12’s Geometry - moment version is a transparent presentation of the necessities of geometry for the highschool scholar. subject matters comprise: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & sector, quantity, and variations. quantity 1 comprises the 1st 6 chapters: fundamentals of Geometry, Reasoning and evidence, Parallel and Perpendicular traces, Triangles and Congruence, Relationships with Triangles, and Polygons and Quadrilaterals.

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8. References and exercises E. Artin. Geometric Algebra, Princeton University Press 1957. H. Baum. Spin-Strukturen and Dirac-Operatoren fiber pseudo-Riemannschen Mannigfaltigkeiten, Teubner Verlag Leipzig 1981. H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors and Killing Spinors on Riemannian Manifolds, Teubner Verlag 1991. 8. References and exercises 33 B. Budinich, A. Trautman. The Spinorial Chessboard, Springer-Verlag 1988. D. Husemoller. Fibre Bundles, McGraw-Hill 1966. B. -L. Michelsohn.

2. Spin Structures 42 Since 7rl (R) contains at most two elements, and n is even, in this case, 7r1(R) = 1. Summarizing, we obtain i (R) 7G2 1 if n is odd, if n is even. Since CIPn is simply connected, we can apply the corresponding criterion for the existence of a spin structure on R. The bundle R has a spin structure if and only if irl (R) = Z2. This leads to the Proposition. n admits a spin structure if and only if n is odd, n - 1 mod 2. Remark. It is important to note that two spin structures on one and the same SO(n)-principal bundle may not be equivalent even if the corresponding Spin(n)-principal bundles over X are equivalent.

1(F)\Spin(n + 1) as well as 7r1(A-1(r)\Spin(n+1)) _ A-' (r). Hence the sequence in question is 1 ) Z2 ) A-1(r) )r ) 1, and the manifold X* = r\Sn is a spin manifold if and only if this sequence splits. For a group F of odd order JFI this sequence always splits. Otherwise, the splitting of the above exact sequence is equivalent to the splitting of the sequence 1 , (A-1(F2) n Z2) ; A-1(F2) > F2 ) 1, where F2 C F is a 2-Sylow subgroup. Summarizing, we obtain the Proposition. Let F C SO(n + 1) be a finite subgroup acting freely on the sphere S.