Showing papers on "Distribution (differential geometry) published in 1976"

On the analytic distributions and foliations of a Kähler manifold

Abstract: A Frobenius-type condition involving d. is proved for the analyticity of a distribution on a complex analytic manifold. AS a consequence, an invariant condition for the analyticity of a distribution on a Kahler manifold is derived and used to establish the local reducibility of some foliate Kahler manifolds with bundle-like metric. The aim of this note is to derive invariant conditions for a distribution of a Kahler manifold to be analytic. These conditions will then be used in the study of some questions concerning complex analytic foliations on Kahler manifolds which, until now, were described only by the use of local coordinates [7]. 1. Let V be a complex analytic manifold of complex dimension n and {zi} (i = 1, ... , n)-local complex coordinates on V. A complex m-dimensional distribution D on V (1 < m < n) is a differentiable subbundle of the complex tangent bundle T(V), with m-dimensional fibres. If we identify T(V) with the bundle which is locally generated by {a/lz1}, it is obvious that the above distribution D is defined locally by (1.1) Wa = a9dz10 (a= 1,...,n-m) where aq are complex valued differentiable functions on V and rank (ar) = n m. Then, the distribution D is said to be analytic if the system (1.1) is equivalent to a similar system with complex analytic coefficients at. This condition is obviously independent of the choice of the local coordinates z'. A first analyticity condition which can be easily established is the following Frobenius-type theorem: 1. THEOREM. The distribution D above is analytic iff (1.2) d. xa = 0 (modulo oa), i.e. (1.3) d coaZ = 7ba A cob (a,b = 1, ... ,n -m), where -7ba are some locally defined diferential forms of type (0, 1). In fact, let (1.1) define an analytic distribution on V. Then we must have Received by the editors December 25, 1974 and, in revised form, November 17, 1975. AMS (MOS) subject classifications (1970). Primary 53C55, 57D30.

Stability of Periodic Linear Systems and the Geometry of Lie Groups

Abstract: Publisher Summary This chapter discusses the stability of periodic linear systems and the geometry of lie groups. In a study on the stability of linear periodic canonical systems, Gel’fand and Lidskii investigate certain aspects of the geometry of the group of symplectic matrices. They show that the interior of the subset of the 2 n x 2 n symplectic matrices ( Sp ( n )), which consists of those matrices similar to orthogonal ones, has 2 n connected components—a particular component being characterized by the distribution of Floquet multipliers of first and second type in the sense of Krein. Krein's work establishes that for Hill's equation, one of these connected components plays a particularly important role. The chapter describes the geometric content of the Liapunov–Krein theorems in the context of the theory of Lie groups. Many results on the stability of Hill's equation have an interpretation in terms of a variational problem of the above type on the symplectic group. The boundedness, asymptotic stability is properties of linear systems of differential equations that are preserved under tensoring and reduction.