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By Shun-Ichi Amari, Hiroshi Nagaoka

ISBN-10: 0821805312

ISBN-13: 9780821805312

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Extra resources for Methods of Information Geometry (Translations of Mathematical Monographs 191)

Example text

D. D2. this local family of disks 342 M, Gromov admits the desired global extension, since the areas of the disks are a priori bounded by const area W and since the transversality of dD2C W to O n T ( W ) prevents the disks from degeneration to cusp-curves (see [Gr02] for more general results). D'2. Examples and corollaries. (a) The above theorem applies to the image WCS3CB4CI~ 2 of the standard sphere S 2 Q S 3 under an arbitrary contact transformation of S 3 (compare [B-G]). D 2. the field O n T ( W ) has not closed orbits, as the curves OD2 ( W cover W\{w o, wl} and they are everywhere transversal to O n T(W).

The group of symplectic diffeomorphisms contracts to the subgroup of isometries. A'~. Let us give a criterion for a closed manifold (V, to) to split as (S 2, eOo)@(S2, COo). We assume the existence of two embedded spheres $1 and $2 with trivial normal bundles in V, such that the form to does not vanish on $1 and on $2 and such that $1 and $2 have a single intersection point where they meet transversally with the intersection number + 1 for the orientations induced by colS1 and t o l S 2. I f every smoothly mapped sphere S ~ V has S to = k ~ to = k ~ to for some integer S $1 $2 k = k ( S ~ V), then the manifold (V, to) splits, (V, to) = (S 2, too)O(S 2, COo).

Insure in the present case holomorphic disks(D2, OD2)~(V, W). ). That is a limit f : (S, ~S)~( V, W) of regular J-curves f j : (S, ~S)~( V, W) is regular unless the map f is nowhere one-to-one on some connected component of S. B]. in case the structure J is complex near W and W is real analytic. Indeed, V lies in a slightly greater manifold V + 3 V, such that a small neighborhood U+CV + of W admits an anti-holomorphic involution I : U + ~ U +. Since OVis J-convex, the intersection VnI(U +n V) equals W.

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Methods of Information Geometry (Translations of Mathematical Monographs 191) by Shun-Ichi Amari, Hiroshi Nagaoka

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