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Non-Compact Riemannian Manifolds with Special Holonomy

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Non-Compact Riemannian Manifolds with Special Holonomy MASARYKOVA UNIVERZITA PRˇ´IRODOVEDECKˇ AFAKULTA´ Yaroslav Bazaykin, PhD, D.Sc. NON-COMPACT RIEMANNIAN MANIFOLDS WITH SPECIAL HOLONOMY habilitaˇcn´ıpr´ace Brno 2019 Abstract This thesis is dedicated to the study of the geometry and topology of non-compact Rieman- nian manifolds with special holonomy groups. After brief review of Riemannian holonomy theory we consider construction of resolution of standard cone over 3-Sasakian manifold. In particular, this gives a continuous family of explicit examples of non-compact Riema- nian manifolds with Sin(7)-holonomy. We apply the idea of this construction to resolve cone G2-holonomy metric over twistor space of 3-Sasakian manifold and to find continu- ous family of SU(4)-holonomy metrics connecting well-known Calabi metrics. The explicit construction of SU(2)-holonomy metric generalising Eguchi-Hanson metrics is studied and applications of these metrics to describing Calabi-Yau metrics on K3 surface are consid- ered. Abstract This thesis is dedicated to the study of the geometry and topology of non-compact Rieman- nian manifolds with special holonomy groups. After brief review of Riemannian holonomy theory we consider construction of resolution of standard cone over 3-Sasakian manifold. In particular, this gives a continuous family of explicit examples of non-compact Riema- nian manifolds with Sin(7)-holonomy. We apply the idea of this construction to resolve cone G2-holonomy metric over twistor space of 3-Sasakian manifold and to find continu- ous family of SU(4)-holonomy metrics connecting well-known Calabi metrics. The explicit construction of SU(2)-holonomy metric generalising Eguchi-Hanson metrics is studied and applications of these metrics to describing Calabi-Yau metrics on K3 surface are consid- ered. Abstract This thesis is dedicated to the study of the geometry and topology of non-compact Rieman- nian manifolds with special holonomy groups. After brief review of Riemannian holonomy theory we consider construction of resolution of standard cone over 3-Sasakian manifold. In particular, this gives a continuous family of explicit examples of non-compact Riema- nian manifolds with Sin(7)-holonomy. We apply the idea of this construction to resolve cone G2-holonomy metric over twistor space of 3-Sasakian manifold and to find continu- ous family of SU(4)-holonomy metrics connecting well-known Calabi metrics. The explicit construction of SU(2)-holonomy metric generalising Eguchi-Hanson metrics is studied and applications of these metrics to describing Calabi-Yau metrics on K3 surface are consid- ered. Contents 1 Introduction 4 References 10 Paper A: Bazaikin, Ya. V., On the new examples of complete noncompact Spin(7)-holonomy metrics. Siberian Mathematical Journal. 2007. Vol. 48, No. 1. P. 8–25. Paper B: Bazaikin, Ya. V., Noncompact Riemannian spaces with the holonomy group Spin(7) and 3-Sasakian manifolds. Proceedings of the Steklov Institute of Mathematics. 2008. Vol. 263, Issue 1. P. 2–12. Paper C: Bazaikin, Ya. V., Malkovich, E. G., Spin(7)-structures on complex linear bundles and explicit Riemannian metrics with holonomy group SU(4). Sbornik Mathematics. 2011. Vol. 202, Issue. 4. P. 467–493. Paper D: Bazaikin, Ya. V., Malkovich, E. G., G(2)-holonomy metrics connected with a 3- Sasakian manifold.. Siberian Mathematical Journal. 2008. Vol. 49, No. 1. P. 1–4. Paper E: Bazaikin, Ya. V., On some Ricci-flat metrics of cohomogeneity two on complex line bundles. Siberian Mathematical Journal. 2004. Vol. 45, No. 3. P. 410–415. Paper F: Bazaikin, Ya. V., Special Kahler metrics on complex line bundles and the geometry of K3-surfaces. Siberian Mathematical Journal. 2005. Vol. 46, No. 6. P. 995–1004. 1 Introduction The first mention of holonomy (namely, the use of the term “holonomic” and “nonholo- nomic” constraints in classical mechanics) dates back to 1895 and belongs to Hertz [28, 38]. In mathematical works, the holonomy concept first appeared in 1923 in E. Cartan papers [15, 16, 18] and already had a modern meaning. Briefly speaking, the holonomy group Hol(M n) of the Riemannian manifold M n is generated by the operators of parallel trans- lations with respect to the Levi-Civita connection along paths starting from and ending at a fixed point p M n. If we consider only contractible loops, we obtain restricted 2 n holonomy group Hol0(M ), which is a connected component of identity map in the group Hol(M n). Further in the text we will assume M n to be simply connected and therefore n n n Hol(M )=Hol0(M ). It is intuitively clear that if Hol(M ) does not coincide with the n maximal possible group of isometries SO(n) of the tangent space TpM , then this should indicate the presence of restrictions on the geometry of the Riemannian manifold. Indeed, each special group of holonomy corresponds to one or another special geometry. The global character of the holonomy group of a Riemannian manifold is emphasized by de Rum decomposition theorem [10]. Namely, it is obvious that if the Riemannian manifold M is a direct product of the Riemannian manifolds M1 and M2,thenHol(M)= Hol(M ) Hol(M ) (together with the corresponding decomposition of the representation 1 ⇥ 2 of the holonomy group). It turns out that if the Riemannian manifold is complete, then the converse is true. The problem of classifying Riemannian holonomy groups naturally arises: which groups can be holonomy groups of a Riemannian manifold? In solving this problem, we can immediately restrict ourselves to complete irreducible Riemannian manifolds, i.e. such whose holonomy representation does not have invariant subspaces in TpM. By the de Rham decomposition theorem, such manifolds do not de- compose into a direct product, and any complete Riemannian manifold decomposes into a product of irreducible ones. Symmetric spaces gives an important example of Riemannian manifolds with special holon- omy groups [18]: Theorem 1. Let M n be a simply connected symmetric space and G be a Lie group of the isometries of M n generated by all transvections. We assume that H G is the isotropy ⇢ group of M, with respect to the chosen point. Then M is isometric to homogeneous 4 space G/H, and the holonomy group Hol(M) coincides with H, and the representation of holonomy coincides with the representation of isotropy. Cartan [17] reduced the problem of describing simply connected Riemannian symmetric spaces to the theory of Lie groups, and he obtained a list of all such spaces. Cartan’s proof and a complete list of simply connected Riemannian symmetric spaces are discussed in [27, 8]. The next important advance in the classification problem was made by Berger: Theorem 1.1. Let M be a simply connected irreducible Riemannian manifold of dimen- sion n that is not symmetric. Then one of the following cases takes place. 1) Hol(M)=SO(n), the general case, 2) n =2m, where m>2 and Hol(M)=U(m) SO(2m), K¨ahler manifolds, ⇢ 3) n =2m, where m>2 and Hol(M)=SU(m) S(2m), special K¨ahler manifolds, ⇢ 4) n =4m, where m>2 and Ho1(M)=Sp(m) SO(4m), hyperk¨ahler manifolds, ⇢ 5) n =4m, where m>2 and Hol(M)=Sp(m)Sp(1) SO(4m), quaternion-K¨ahler ⇢ manifolds, 6) n =7and Hol(M)=G SO(7), 2 ⇢ 7) n =8and Ho1(M)=Spin(7) SO(8). ⇢ In the original Berger list there was also the case n = 16 and Hol(M)=Spin(9) SO(16). ⇢ However, in [7, 11] it was proved that in this case M is a symmetric and is isometrical to 2 projective Cayley plane CaP = F4/Spin(9). Thus, to solve the classification problem, one needs to understand which of the groups in the Berger list can be realized as holonomy groups of complete Riemannian manifolds. In this case, two aspects of the classification problem arise: the proof that the Berger group is realized as the holonomy group of the (incomplete) locally defined Riemannian metric; finding the complete Riemannian metric with a given holonomy group. The second problem, especially in the case of constructing a Riemannian metric on a closed manifold, is significantly more difficult. On the other hand, the construction of a complete metric seems to be a reasonable problem, due to the global nature of the holonomy group (we cannot potentially exclude the possibility that loops that can go “far enough” from a fixed point will have a decisive influence on the holonomy group). Furher we briefly go through the Berger list and comment each case. 5 - K¨ahler spaces are well studied, and many examples of K¨ahler spaces with the holonomy group U(m) can be cited [26, 34]. - Riemannian manifolds whose holonomy group is contained in SU(m) are called Calabi – Yau manifolds (the name is connected with the Calabi – Yau theorem, cited below), or special K¨ahler manifolds. It can be shown that special K¨ahler manifolds are Ricci-flat [34, 8]. Already from this fact it is clear that the construction of such manifolds is a difficult task. The first example of a complete Riemannian metric with the group SU(m) was constructed by Calabi [14]: d⇢2 1 ds˜2 = + ⇢2 1 (d⌧ 2A)2 + ⇢2ds2. 1 1 − ⇢2n − − ⇢2n ✓ ◆ 2 n 1 Here ds is the Fubini – Study metric on the complex projective space CP − , A is the n 1 1-form integrating the K¨ahler form on CP − , ⇢ 1, and ⌧ is the angular variable on the ≥ circle. It can be shown that metric ds˜2 is a smooth globally defined special K¨ahler metric n 1 of the n-th tensor degree of the complex Hopf line bundle over CP − . Note the Calabi metric in such a form was found in [37], a similar approach to the construction of this metric was used in [9].
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