By Luther Pfahler Eisenhart
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Extra resources for An Introduction To Differential Geometry With Use Of Tensor Calculus
2. Let (M, σ) be a compact connected symplectic manifold equipped with a free symplectic action of a torus T for which at least one, and hence every of its T -orbits is a dim T -dimensional symplectic submanifold of (M, σ), and let µh : H1 (M/T, Z) → T be the homomorphism induced on homology via the Hurewicz map by the monodromy homomorphism µ from π1 (M/T, p0 ) into T (cf. 1)). e. µh satisﬁes that µh ([γ]) = 1 for every [γ] ∈ H1 (M/T, Z) of ﬁnite order. Proof. 2) φD : M/T ×π1 (M/T, p0 ) T → M.
7. Let (M, σ) be a compact connected 2n-dimensional symplectic manifold equipped with a free symplectic action of a (2n − 2)-dimensional torus T , for which at least one, and hence every T -orbit is a symplectic submanifold of (M, σ). If αi , βi and αi , βi , 1 ≤ i ≤ g, are symplectic bases of H1 (M/T, Z), then for every homomorphism f : H1 (M/T, Z) → T , we have that the Sp(2g, Z)-orbits of the tuples (f (αi ), f (βi ))gi=1 and (f (αi ), f (βi ))gi=1 are equal. 8. Let T be a torus. Let g be a non-negative integer, and let Σ, Σ be compact, connected, oriented, smooth surfaces of the same genus g.
3) 1 We Σ /T → Σ/π1 (Σ, p0 ) → Σ, Mmodel Σ Σ should probably write Mmodel instead of Mmodel , but we avoid to write the de, p0 pendance on p0 to shorten the notation, and since the models are identiﬁed for all choices of p0 . 40 5. 4) ΩΣ µh model = µh ◦ f. 4 f preserves the intersection form and hence the unique collection of elements αi , βi , 1 ≤ i ≤ g such that f (αi ) = αi and f (βi ) = βi , for all 1 ≤ i ≤ g, is a symplectic basis of the homology ΩΣ ΩΣ Σ /T, Z). Let γ be the 2g-tuple of elements µh model (αi ), µh model (βi ), group H1 (Mmodel 1 ≤ i ≤ g.
An Introduction To Differential Geometry With Use Of Tensor Calculus by Luther Pfahler Eisenhart