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By Greene R., Yau S.-T. (eds.)

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Extra resources for Differential Geometry: Geometry in Mathematical Physics and Related Topics, Part 2

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S~1) (~)nr(~)=p. If IT is a pro- Proof. Clearly, we have jective plane (n=2), then every line m, say, passing through P is a 1-chord of ~, and m#S(1) (~) implies P#mnmEr(~). P If n~3, then (PQ)~ -1 = ((PQ)~ -1 S~ 2 ) (~) C 1 ns~ 2) (~) vQ)n((PQ)~ -1-1 vQ)~ , from which it follows that S~1) (,) is a 1-chord of ,. On the other hand, assume that m=LnL~ is any 1-chord of ~, where L denotes a plane passing through P. 4) r;1 n~3, (UII/S~1) the restricted map (r;» : UII/S~1) (r;) - UII/PQ is a non-degenerate projective isomorphism in the (n-1)dimensional quotient space II/P.

4. In rr=rr(v) we choose two different points P and Q. 2) r; : urr/p - urr/Q be a projective isomorphism. 3) r(r;) := {XEPjXEl and XElr; for some line 13P} as being the point set generated by r;. is a generating map of r(r;). cause P,QEPQ and PE(PQ)r; -1 Obvio~sly p,QEr(r;), be- , QE(PQ)r;. The fundamental sub- space of r; is defined as the intersection of all r;-invariant subspaces and is denoted by G(r;). If r;' :urr/p' - urr/Q' is any generating map of r(r;), then {P' ,Q'} is called a fundamental pair of the point set r(r;) and we shall use the term fundamental point of r(r;) for p' as well as Q'.

The subspaces of n form the lattice (un,v,n) with v and n denoting the operation signs for "join" and "intersection", respectively. 2. 41 POLYNOMIAL IDENTITIES IN DESARGUESIAN PROJECTIVE SPACES u(rr(M}} = urr(M} = {xEurrlxcM}, u(rr/M} = urr/M = {xEurrlx~M}, respectively. We shall not distinguish a point MEP from the subspace {M}Eurr and urr/M will be called a bundLe (of sub- spaces). The same symbol will denote a coLLineation (being a point-to-point map) and the associated isomorphism (which maps subspaces to subspaces).

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