A differentiable manifold is said to be contact if it admits a linear functional f on the tangent bundle satisfying f ∧(df)^(M-1)≠0.This remark obtain the following the classification:Let M be a complete connected c...A differentiable manifold is said to be contact if it admits a linear functional f on the tangent bundle satisfying f ∧(df)^(M-1)≠0.This remark obtain the following the classification:Let M be a complete connected contact hyper-surface of CH^2(-4),then M is congruent to one of the following: (i)A tube of radius r>0 around a totally geodesic,totally real hyperbolic space form H^2(-1); (ii)A tube of radius r>0 around a totally geodesic complex hyperbolic space form CH^1(-4); (iii)A geodesic hypersphere of radius r>0,or (iv)A horosphere.展开更多
Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgr...Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.展开更多
The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×...The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.展开更多
文摘A differentiable manifold is said to be contact if it admits a linear functional f on the tangent bundle satisfying f ∧(df)^(M-1)≠0.This remark obtain the following the classification:Let M be a complete connected contact hyper-surface of CH^2(-4),then M is congruent to one of the following: (i)A tube of radius r>0 around a totally geodesic,totally real hyperbolic space form H^2(-1); (ii)A tube of radius r>0 around a totally geodesic complex hyperbolic space form CH^1(-4); (iii)A geodesic hypersphere of radius r>0,or (iv)A horosphere.
文摘Let X denote a complex analytic manifold, and let Aut(X) denote the space of invertible maps of a germ (X, a) to a germ (X, b); this space is obviously a groupoid; roughly speaking, a 'Lie groupoid' is a subgroupoid of Aut(X) defined by a system of partial differential equations.To a foliation with singularities on X one attaches such a groupoid, e.g. the smallest one whose Lie algebra contains the vector fields tangent to the foliation. It is called 'the Galois groupoid of the foliation'. Some examples are considered, for instance foliations of codimension one, and foliations defined by linear differential equations; in this last case one recuperates the usual differential Galois group.
基金supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(Grant No.2013004848)
文摘The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.
