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Let M be a small Seifert fiber space which has also a structure of surface bundle F x [0, 1]/{(x, 0) = (f(x), 1)} over the circle, where f: F -> F is a monodromy map with non-empty fixed point set. A typical example of such a manifold appears as the result of 0-surgery on a torus knot. For each section in M, we have a ‘projection’ in F in a natural way. We give a condition assuring that the given section in M is hyperbolic in terms of the ‘projection’ in the fiber surface. By translating the result, we give a condition to obtain pseudo-Anosov automorphisms of once punctured surfaces from a periodic automorphism.
Source:HYPERBOLIC SECTIONS IN SEIFERT-FIBERED SURFACE BUNDLES
Nov 09, 2009 at 5:00 pm.
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This is a foundational paper on flows of G2-structures. We use local coordinates to describe the four torsion forms of a G2-structure and derive the evolution equations for a general flow of a G2-structure on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray.
As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G2-geometry.
Source:FLOWS OF G2-STRUCTURES, I
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