A History of Geometrical Methods by Julian Lowell Coolidge

By Julian Lowell Coolidge

Full, authoritative background of the ideas for facing geometric equations covers improvement of projective geometry from historic to fashionable occasions, explaining the unique works, commenting at the correctness and directness of proofs, and displaying the relationships among arithmetic and different highbrow advancements. 1940 edition.

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E. in Ω, together imply that ∇Φ(x) ∈ O3+ for almost all x ∈ U . By (i), there thus exist c ∈ E3 and Q ∈ O3+ such that Φ(x) = Θ(x) = c + Q ox for almost all x = Θ(x) ∈ U , or equivalently, such that Ξ(x) := ∇Θ(x)∇Θ(x)−1 = Q for almost all x ∈ U. Since the point x0 ∈ Ω is arbitrary, this relation shows that Ξ ∈ L1loc (Ω). By a classical result from distribution theory (cf. 6]), Sect. 8] Uniqueness of immersions with the same metric tensor 37 we conclude from the assumed connectedness of Ω that Ξ(x) = Q for almost all x ∈ Ω, and consequently that Θ(x) = c + QΘ(x) for almost all x ∈ Ω.

Let Ω be a connected and simply-connected open subset of n n ) ∈ C 2 (Ω; S3> ), n ≥ 0, be matrix fields satisfying Rqijk = 0 in R3 . Let Cn = (gij Ω, n ≥ 0, such that lim n→∞ Cn − I 2,K = 0 for all K Ω. Then there exist immersions Θn ∈ C 3 (Ω; E3 ) satisfying (∇Θn )T ∇Θn = Cn in Ω, n ≥ 0, such that lim n→∞ Θn − id 3,K = 0 for all K Ω where id denotes the identity mapping of the set Ω, the space R3 being identified here with E3 (in other words, id(x) = x for all x ∈ Ω). Three-dimensional differential geometry 40 [Ch.

7-1), of which Liouville’s theorem is the special case corresponding to Θ(x) = x for all x ∈ Ω. (3) The result established in part (i) of the above proof asserts that, given a connected open subset Ω of R3 , if a mapping Θ ∈ H 1 (Ω; E3 ) is such that ∇Θ(x) ∈ O3+ for almost all x ∈ Ω, then there exist c ∈ E3 and Q ∈ O3+ such that Θ(x) = c + Qox for almost all x ∈ Ω. This result thus constitutes a generalization of Liouville’s theorem. 7-3), an assumption about the sign of det ∇Θ becomes necessary.

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