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On Geometry of Manifolds with Some Tensor Structures and Metrics Of University of Plovdiv Paisii Hilendarski Faculty of Mathematics and Informatics Department of Algebra and Geometry Mancho Hristov Manev On Geometry of Manifolds with Some Tensor Structures and Metrics of Norden Type Dissertation Submitted for the Scientific Degree: Doctor of Science Area of Higher Education: 4. Natural Sciences, Mathematics and Informatics Professional Stream: 4.5. Mathematics Scientific Specialty: Geometry and Topology arXiv:1706.05505v1 [math.DG] 17 Jun 2017 Plovdiv, 2017 To my wife Rositsa 2 Structure of the Dissertation Structure of the Dissertation The present dissertation consists of an introduction, a main body, a conclusion and a bibliography. The introduction consists of two parts: a scope of the topic and the purpose of the dissertation. The main body includes two chapters containing 15 sections. The conclusion provides a brief summary of the main contributions of the dissertation, a list of the publications on the results given in the dissertation, a declaration of originality and acknowledgements. The bibliography contains a list of 155 publications used in the text. >>> 3 Contents Contents Introduction 6 Scope of the Topic .............................................. 6 Purpose of the Dissertation ..................................... 9 Chapter I. On manifolds with almost complex structures and almost contact structures, equipped with metrics of Norden type 11 §1. Almost complex manifolds with Norden metric .......... .... 12 §2. Invariant tensors under the twin interchange of Norden metrics on almost complex manifolds .................... .... 22 §3. Canonical-type connections on almost complex manifolds with Norden metric ................................... ...... 40 §4. Almost contact manifolds with B-metric .............. ....... 49 §5. Canonical-type connection on almost contact manifolds with B-metric ....................................... ........ 62 §6. Classification of affine connections on almost contact manifolds with B-metric .............................. ...... 78 §7. Pair of associated Schouten-van Kampen connections adapted to an almost contact B-metric structure ................. .... 96 §8. Sasaki-like almost contact complex Riemannian manifolds ... 108 4 Contents Chapter II. On manifolds with almost hypercomplex structures and almost contact 3-structures, equipped with metrics of Hermitian-Norden type 129 §9. Almost hypercomplex manifolds with Hermitian-Norden metrics ............................................ ......... 130 §10. Hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie algebras ......................... ..... 135 §11. Tangent bundles with complete lift of the base metric and almost hypercomplex Hermitian-Norden structure . .. 144 §12. Associated Nijenhuis tensors on almost hypercomplex manifolds with Hermitian-Norden metrics ............... ... 155 §13. Quaternionic Kähler manifolds with Hermitian-Norden metrics ............................................ ........ 170 §14. Manifolds with almost contact 3-structure and metrics of Hermitian-Norden type ............................... ..... 180 §15. Associated Nijenhuis tensors on manifolds with almost contact 3-structure and metrics of Hermitian-Norden type . 196 Conclusion 208 Contributions of the Dissertation ............................... 208 Publications on the Dissertation ................................ 211 Declaration of Originality ...................................... 213 Acknowledgements .............................................. 214 Bibliography 215 >>> 5 Introduction. Scope of the Topic Introduction Scope of the Topic Among additional tensor structures on a smooth manifold, one of the most studied is almost complex structure, i.e. an endomorphism of the tan- gent bundle whose square, at each point, is minus the identity. The mani- fold must be even-dimensional, i.e. dim = 2n. Usually it is equipped with a Hermitian metric which is a (Riemannian or pseudo-Riemannian) met- ric that preserves the almost complex structure, i.e. the almost complex structure acts as an isometry with respect to the (pseudo-)Riemannian metric. The associated (0,2)-tensor of the Hermitian metric is a 2-form and hence the relationship with symplectic geometry. A relevant counterpart is the case when the almost complex struc- ture acts as an anti-isometry regarding a pseudo-Riemannian metric. Such a metric is known as an anti-Hermitian metric or a Norden met- ric (first studied by and named after A. P. Norden [120, 121]). The Nor- den metric is necessary pseudo-Riemannian of neutral signature whereas the Hermitian metric can be Riemannian or pseudo-Riemannian of sig- nature (2n1, 2n2), n1 + n2 = n. The associated (0,2)-tensor of any Nor- den metric is also a Norden metric. So, in this case we dispose with a pair of mutually associated Norden metrics, known also as twin Nor- den metrics. This manifold can be considered as an n-dimensional mani- fold with a complex Riemannian metric whose real and imaginary parts are the twin Norden metrics. Such a manifold is known as a gener- alized B-manifold [49, 108, 50, 45], an almost complex manifold with Norden metric [147, 35, 34, 14, 23, 123, 124, 131], an almost complex manifold with B-metric [36, 38], an almost complex manifold with anti- Hermitian metric [16, 17, 28] or a manifold with complex Riemannian metric [65, 105, 37, 64]. Supposing a manifold is of an odd dimension, i.e. dim = 2n +1, then 6 Introduction. Scope of the Topic there exists a contact structure. The codimension one contact distribution can be considered as the horizontal distribution of the sub-Riemannian manifold. This distribution allows an almost complex structure which is the restriction of a contact endomorphism on the contact distribution. The vertical distribution is spanned by the corresponding Reeb vector field. Then the odd-dimensional manifold is equipped with an almost contact structure. If we dispose of a Hermitian metric on the contact distribution then the almost contact manifold is called an almost contact metric mani- fold. In another case, when a Norden metric is available on the con- tact distribution then we have an almost contact manifold with B-metric. Any B-metric as an odd-dimensional counterpart of a Norden metric is a pseudo-Riemannian metric of signature (n +1,n). A natural generalization of almost complex structure is almost hyper- complex structure. An almost hypercomplex manifold is a manifold which tangent bundle equipped with an action by the algebra of quaternions in such a way that the unit quaternions define almost complex structures. Then an almost hypercomplex structure on a 4n-dimensional manifold is a triad of anti-commuting almost complex structures whose triple com- position is minus the identity. It is known that, if the almost hypercomplex manifold is equipped with a Hermitian metric, the derived metric structure is a hyper-Hermitian structure. It consists of the given Hermitian metric with respect to the three almost complex structures and the three associated Kähler forms [2]. Almost hypercomplex structures (under the terminology of C. Ehresmann of almost quaternionic structures) with Hermitian metrics were studied in many works, e.g. [29, 122, 13, 150, 134, 18, 3]. An object of our interest is a metric structure on the almost hyper- complex manifold derived by a Norden metric. Then the existence of a Norden metric with respect to one of the three almost complex structures implies the existence of one more Norden metric and a Hermitian metric with respect to the other two almost complex structures. Such a metric is called a Hermitian-Norden metric on an almost hypercomplex manifold. Furthermore, the derived metric structure contains the given metric and three (0,2)-tensors associated by the almost hypercomplex structure – a Kähler form and two Hermitian-Norden metrics for which the roles of the almost complex structures change cyclically. Thus, the derived manifold is called an almost hypercomplex manifold with Hermitian-Norden met- 7 Introduction. Scope of the Topic rics. Manifolds of this type are studied in several papers with the author participation [47, 81, 103, 46, 82, 97]. The notion of almost contact 3-structure is introduced by Y. Y. Kuo in [62] and independently under the name almost coquaternion structure by C. Udrişte in [146]. Later, it is studied by several authors, e.g. [62, 63, 137, 153]. It is well known that the product of a manifold with almost contact 3-structure and a real line admits an almost hypercomplex structure [62, 2]. All authors have previously considered the case when there exists a Riemannian metric compatible with each of the three structures in the given almost contact 3-structure. Then the object of study is the so-called almost contact metric 3-structure (see also [5, 21, 19, 20]). Compatibility of an almost contact 3-structure with a B-metric is not yet considered before the publications that are part of this dissertation. In the present work we launch such a metric structure on a manifold with almost contact 3-structure. >>> 8 Introduction. Purpose of the Dissertation Purpose of the Dissertation The object of study in the present dissertation are some topics in dif- ferential geometry of smooth manifolds with additional tensor structures and metrics of Norden type. There are considered four cases depending on dimension of the manifold: 2n, 2n +1, 4n and 4n +3. The studied tensor structures, which are counterparts in the different related dimen- sions, are: the almost complex structure, the almost contact
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