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These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class C 1. We conclude with some remarks about the stationary measure.

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Popular Searches simpsons and mathematics 11 and 12 math textbook concept based mathematics all you wanted to know about mathematics precalculus mathematics for calculus. Item Added: Integrable Systems and Foliations. View Wishlist. De Rham, [71J, p. Let now a be an arbitrary top degree form on M. Then g1 is a 1. We have proved the following cf.

## Progress in Mathematics

Lichnerowicz, [] : Theorem 2. Chapter 3 Examples 3. Nomizu, , vol. Hence the result in [81 is wrong. Endowing R x Sii-1 with the standard product metric which is Z-invariant, one may check that f is an isometry. As we shall see later on, this observation leads to an important generalization of Hopf manifolds, namely to the class of generalized Hopf manifolds. It may be shown that it fails to be parallel with respect to the Levi-Civita connection of gt. We should mention that no attempt to construct l. The existence of l.

The metrics gt appear for the first time in []. To construct them we start cf. Let al, a2, a3 t be a real eigenvector corresponding to a and b1,b2ib3 t an eigenvector for 0. We wish to compute its Betti numbers. The generators 3.

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The elements dependent over Z by the properties of M. Thus Sm admits no global Kahler metrics. This will be referred to as the Tricerri metric. Thus 3. Note that 3. We shall now follow F. Tricerri [ to compute the covariant derivative of w. As our considerations are local in character, we may as well perform all calculations on H x C. Thus DE; B 0 for j 34 0 so that the Lee form is not parallel, yet it may be easily seen to be harmonic. However, it is not regular because the Inoue surfaces have no complex curves.

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Finally, we mention that other Inoue surfaces possess natural l. Tricerri, []. Let r r, 1 C H r, 1 be the subgroup of all matrices with integer entries. This is a compact manifold. The 1-forms, vector fields, and metric constructed above project on N r, 1 via the natural N r, 1. For simplicity we shall denote their images by projection H r, 1 the same symbols. De Leon, [68] with respect to the natural complex structure. Thus J is a complex structure.

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Mencia, [6]. This example provides a family of compact l. We first describe the construction of M3 k , cf. Hahn, [9].

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## Homogénéité locale pour les métriques riemanniennes holomorphes en dimension $3$

By a result in [9] there exist discrete subgroups of G k such that the corresponding quotient is compact. Let a,,3, -y be the projections on M3 k of the 1-forms in the above basis of right invariant forms on G k. As the circle bundles over M3 k are classified by H2 M3 k , Z , for each integer n there is a principal circle bundle it : M4 k,n , M3 k associated to nA[a A,3].

Moreover there is a connection 1-form 77 whose curvature form is nAa A,3. Thus M4 k, n admits no global Kahler metrics. We build a l. Then J is integrable and g is Hermitian. Moreover Vw 3k 0 so that M4 k,n is l. In [6] an explicit realization of M4 k,n is also given. Gauduchon for the following example. With this structure, S is called a Berger sphere.

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Vaisman constructed cf. Rocha have recently constructed new examples in []. Then one may check that the resulting structure is l. However, it seems to be difficult to find noncompact examples which admit no global Kahler metrics. Van de Ven, [45[, have generalized complex Hopf manifolds as follows.

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Then Z acts freely and properly discontinuously on B2n b as a subgroup of C. Note that H 1 ,. Let D1 be the punctured open unit disk in C. The following result holds cf. For any a E D1 there is a diffeomorphism between it-1 a and HQ b. Then each fibre 7r-1 a , a E M, is a compact complex manifold.

Note that the action 3.