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... for any Z ∈ p [8]. In [3] it is proved the following (compare with [26, Theorem 4.1]) ...

... The discussion in Section 7 implies that any G 2 -invariant metric on these spaces, whose full connected isometry group is not G 2 , is SO(7)-normal (symmetric) metric on G 2 /SU(2) · SO(2) = Gr + 7,2 . The last two closed symmetric root subsystems are maximal, so they correspond to maximal Lie subalgebras in g 2 , which are respectively isomorphic to su(2) ⊕ su (2) and su (3) with the corresponding compact connected Lie subgroups SO (4) and SU(3) and homogeneous spaces G 2 /SO(4) and G 2 /SU(3) = S 6 , compare with [19]. Note that G 2 /SO(4) is irreducible symmetric space, see [8]. ...

... Let G be another simple group (with roots of different lengths), and α, β ∈ Δ l . Then α, β = 0 or (α, β) = 2π 3 or (α, β) = π 3 . In the first case α ± β cannot be a roots, so [V α , V β ] = 0. ...

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ-homogeneous spaces in the case of Riemannian manifolds and prove that they constitute a new proper subclass of geodesic orbit (g.o.) spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian spaces.

... We also remark that the Reeb vector field ξ is characterized uniquely by the properties η(ξ ) = 1, dη(ξ, · ) = 0 and that the cone over a Sasakian manifold admits a Kähler structure. 2,3 such that each (M, g, ξ i , η i , ϕ i ) is a Sasakian structure and g(ξ i , ξ j ) = δ i j , [ξ i , ξ j ] = 2ε i jk ξ k , where ε i jk denotes the Levi-Civita symbol and (i, j, k) is a permutation of (1, 2, 3). ...

... As we will show in this article, the simply connected homogeneous 3-Sasakian manifolds are of the form G/(C G (K )) 0 , where (C G (K )) 0 is the identity component of the centralizer C G (K ) of K in G. In 1968 Alekseevskii fully classified compact homogeneous positive qK manifolds by demonstrating that they are necessarily of the form G/N G (K ) [2,Theorem 1]. ...

We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer et al. (J Reine Angew Math 455:183–220, [10]). In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss why the real projective spaces are the only non-simply connected homogeneous 3-Sasakian manifolds and derive the famous classification of homogeneous positive quaternionic Kähler manifolds due to Alekseevskii (Funct Anal Appl 2(2):106–114, [2]) from our results.

... For an arbitrary sequence of monopoles {(A n , u n )} ∞ n=1 there may be no Y (X ) as in Theorem 1 (2). For example, if lim n→∞ ρ 0 • u n = 0, then we cannot find such submanifolds Y and X . ...

... The Wolf spaces are the only compact, homogeneous, quaternionic Kähler manifolds. They are classified by Wolf [19] and Alekseevskii [1,2]. Some examples of Wolf spaces are as follows: ...

The purpose of this paper is to study the behaviour of sequences of generalised monopoles with a uniform bound on a certain L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm. We focus on the case that the target hyperKähler manifolds are Swann bundles. In 3-dimensional case, suppose that there exists an open submanifold Y′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y'$$\end{document} such that the hyperKähler potential along the monopoles has a uniform lower bound over Y′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y'$$\end{document}. Then we show that there exist convergent subsequences of generalised monopoles over any compact subset of Y′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y'$$\end{document}. Under similar assumptions, the same conclusion holds for the generalised harmonic spinors in dimension four.

... As for the vector multiplets, only symmetric spaces will be relevant for consistent truncations. The Riemannian symmetric quaternionic-Kähler spaces were first considered by Wolf in [40] and then classified by Alekseevsky in [41]. This was then extended to the JHEP06(2022)003 pseudo-Riemannian class by Alekseevsky and Cortés in [42]. ...

... , (4.104) where G H = C E 6(6) (G S ) is the group generated by the singlets J A . The above considerations already restrict the possible scalar manifolds for the hypermultiplets to the following list [41,42] M H = F 4(4) SU(2) · USp (6) , n H = 7 , ...

A bstract
Using exceptional generalised geometry, we classify which five-dimensional $$ \mathcal{N} $$ N = 2 gauged supergravities can arise as a consistent truncation of 10-/11-dimensional supergravity. Exceptional generalised geometry turns the classification into an algebraic problem of finding subgroups G S ⊂ USp(8) ⊂ E 6(6) that preserve exactly two spinors. Moreover, the intrinsic torsion of the G S structure must contain only constant singlets under G S , and these, in turn, determine the gauging of the five-dimensional theory. The resulting five-dimensional theories are strongly constrained: their scalar manifolds are necessarily symmetric spaces and only a small number of matter multiplets can be kept, which we completely enumerate. We also determine the largest reductive and compact gaugings that can arise from consistent truncations.

... As for the vector multiplets, only symmetric spaces will be relevant for consistent truncations. The Riemannian symmetric quaternionic-Kähler spaces were first considered by Wolf in [39] and then classified by Alekseevsky in [40]. This was then extended to the pseudo-Riemannian class by Alekseevsky and Cortés in [41]. ...

... The above considerations already restrict the possible scalar manifolds for the hyper-multiplets to the following list [40,41] M H = F 4(4) SU(2) · USp (6) , n H = 7 , ...

Using exceptional generalised geometry, we classify which five-dimensional ${\cal N}=2$ gauged supergravities can arise as a consistent truncation of 10-/11-dimensional supergravity. Exceptional generalised geometry turns the classification into an algebraic problem of finding subgroups $G_S \subset \mathrm{USp}(8) \subset \mathrm{E}_{6(6)}$ that preserve exactly two spinors. Moreover, the intrinsic torsion of the $G_S$ structure must contain only constant singlets under $G_S$, and these, in turn, determine the gauging of the five-dimensional theory. The resulting five-dimensional theories are strongly constrained: their scalar manifolds are necessarily symmetric spaces and only a small number of matter multiplets can be kept, which we completely enumerate. We also determine the largest reductive and compact gaugings that can arise from consistent truncations.

... So φ 1 φ QX = QX , that is, φQX = −φ 1 QX . Then we have our assertion (1). ...

... Thus B is a quaternionic and real submanifold of G * 2 ރ( m+2 ). Since B is quaternionic, it is totally geodesic in G * 2 ރ( m+2 ) (see Alekseevskiȋ [1]). The only quaternionic totally geodesic submanifolds of G * 2 ރ( m+2 ), m = 2n ≥ 4, of half dimension are G * 2 ރ( n+2 ) and ވH n (see Berndt [2]). ...

... It is the quaternionic maximal subbundle of T p M = Q ⊕ Q ⊥ . By using the result of Alekseevskii [1], Berndt and Suh [3] have classified all real hypersurfaces with the invariant properties in G 2 (C m+2 ) as follows: ...

... In Sections 1 and 2, we give a complete proof of Theorems 1 and 2, respectively. We refer to [1], [3], [4] and [9] for Riemannian geometric structures of G 2 (C m+2 ), m ≥ 3. ...

In this paper, we have considered new restricted commuting con-ditions between the restricted Jacobi operator and the Ricci tensor for real hypersurfaces in complex two-plane Grassmannians G2(Cm+2). By using a new method of simultaneous diagonalzation for commuting symmetric opera-tors, we veried a complete classification for real hypersurfaces in G2(Cm+2) with above conditions that become equivalent condition for hypersurfaces of Type A. © 2015 University of Houston.ed a complete classification for real hypersurfaces in G2(Cm+2) with above conditions that become equivalent condition for hypersurfaces of Type A.

... where the action of the quaternions H * mods out by SU(2) × R + . The Riemannian symmetric quaternionic-Kähler spaces were first considered by Wolf in [83] and classified by Alekseevsky in [84], while the pseudo-Riemannian case was analysed by Alekseevsky and Cortés [85], and (3.23) is indeed included in their list. Recall that one can always construct a hyper- Kähler cone, known as the Swann bundle, over any quaternionic-Kähler space [86]. ...

... ) (as always we are ignoring discrete factors). The Riemannian case was first studied by Wolf in [83] and classified by Alekseevsky in [84], while the pseudo-Riemannian case, of relevance here, was analysed by Alekseevsky and Cortés in [85]. It is known that every quaternionic-Kähler manifold admits a bundle over it whose structure group is SU(2) [149]. ...

In this paper we define the analogue of Calabi--Yau geometry for generic
$D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory.
We show that solutions of the Killing spinor equations are in one-to-one
correspondence with integrable, globally defined structures in
$E_{7(7)}\times\mathbb{R}^+$ generalised geometry. Such "exceptional
Calabi--Yau" geometries are determined by two generalised objects that
parametrise hyper- and vector-multiplet degrees of freedom and generalise
conventional complex, symplectic and hyper-Kahler geometries. The integrability
conditions for both hyper- and vector-multiplet structures are given by the
vanishing of moment maps for the "generalised diffeomorphism group" of
diffeomorphisms combined with gauge transformations. We give a number of
explicit examples and discuss the structure of the moduli spaces of solutions.
We then extend our construction to $D=5$ and $D=6$ flux backgrounds preserving
eight supercharges, where similar structures appear, and finally discuss the
analogous structures in $O(d,d)\times\mathbb{R}^+$ generalised geometry.

... By using the result in Alekseevskii [1], Berndt and Suh [3] proved the following result about space of type (A)(sentence about (A)) and type (B)(one about (B)) : ...

... In section 2, we give a complete proof of the main theorem and corollary, respectively. In this paper, we refer to [1,3,4,7,11] for Riemannian geometric structures of G 2 (C m+2 ) and its geometric quantities, respectively. ...

Using generalized Tanaka-Webster connection, we considered a real
hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$
when the GTW Reeb Lie derivative of the structure Jacobi operator coincides
with the Reeb Lie derivative. Next using the method of simultaneous
diagonalization, we prove a complete classification for a real hypersurface in
$G_2({\mathbb C}^{m+2})$ satisfying such a condition. In this case, we have
proved that $M$ is an open part of a tube around a totally geodesic
$G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

... Moreover, the corresponding geometrical feature for hypersurfaces in HP m is the invariance of the distribution D ⊥ = Span{ξ 1 , ξ 2 , ξ 3 } by the shape operator, where ξ i = −J i N , J i ∈ J. In fact, every tube around a quaternionic submanifold of HP m admits a geometrical structure of this type (see Alekseevskii [1]). ...

... From such a view point, we considered two natural geometric conditions for real hypersurfaces in G 2 (C m+2 ) that [ξ] = Span{ξ} and D ⊥ = Span{ξ 1 , ξ 2 , ξ 3 } are invariant under the shape operator. By using such conditions and the result in Alekseevskii [1], Berndt and Suh [2] have proved the following: The structure vector field ξ of a real hypersurface M in G 2 (C m+2 ) is said to be a Reeb vector field. If the Reeb vector field ξ of a real hypersurface M in G 2 (C m+2 ) is invariant by the shape operator, M is said to be a Hopf hypersurface. ...

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.

... In the proof of Theorem 1 we have proved that the one-dimensional distribution [ξ] is contained in either the 3-dimensional distribution D ⊥ or in the orthogonal complement D such that T x M = D⊕D ⊥ . The case (2) in Theorem A is just the case that the one dimensional distribution [ξ] is contained in D ⊥ . Of course it is not difficult to check that the second fundamental form of any real hypersurfaces given in Theorem 1 is not parallel. ...

... The corresponding geometrical feature is that the three dimensional distribution J(⊥M ) on M , which is obtained by applying the almost Hermitian structures in the quaternionic Kähler structure J of HP m to ⊥M , is invariant under A. In fact, every tube around a quaternionic submanifold of HP m has this geometrical feature. (Note that by a result of D.V. Alekseevskii[2] such a quaternionic submanifold is necessarily totally geodesic.) But the converse is not true. ...

In this paper the present author introduce some recent results about real hyper-surfaces in complex two-plane Grassmannians G2(C m+2) , which give an extended version of the talk delivered at the 8th International Workshop on Differential Geometry held at Kyungpook National University during the period from 12 December to 13 December 2003.

... and AQ ⊥ ⊂ Q ⊥ . By using the result of Alekseevskii [1], Berndt and Suh [2] have classified all real hypersurfaces with two natural invariant properties in G 2 (C m+2 ) as follows: ...

... In this paper, we refer [1], [2], [3], [10] and [7], [14], [15] for Riemannian geometric structures of G 2 (C m+2 ) and its geometric quantities, respectively. ...

In this paper, we introduce new notions of semi-parallel shape operators and
structure Jacobi operators in complex two-plane Grassmannians $G_2({\mathbb
C}^{m+2})$. By using such a semi-parallel condition, we give a complete
classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.

... In Sec-tion 4, we show that they comprise the center of the automorphism algebra and conclude that they generate a closed subgroup of the automorphism group. Combining this with classical results due to Alekseevskii [4] as well as Conner and Raymond [9], we prove that the only connected compact homogeneous degenerate 3-(α, δ)-Sasakian manifold is the trivial example of the 3-torus T 3 (Section 5). Finally, we study simply connected nilpotent Lie groups with a left-invariant degenerate 3-(α, δ)-Sasakian structure in Section 6. Again using traditional results by Wilson [18] as well as Milnor [16], we show there is exactly one family of such spaces, the quaternionic Heisenberg groups. ...

We propose a new method to construct degenerate 3-(α, δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α, δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

... For example [4], when (G, H) are (E 6 , SU (6)), (E 7 , SO (12)), or (E 8 , E 7 ), half-hypermultiplets appear in 20, 32, or 56 of the respective H. They are all pseudo-real representations and correspond, not to homogeneous Kähler manifolds, but to quaternionic Kähler symmetric spaces known as Wolf spaces [12][13][14][15] (see Ref. [16] for a review): ...

In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic Kähler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by the intriguing singularity structure previously found in such F-theory models with a gauge group SU(6), SO(12) or E7, we investigate, as the final magical example, an F-theory on an elliptic fibration over a Hirzebruch surface of the non-split I6 type, in which the unbroken gauge symmetry is supposed to be Sp(3). We find significant qualitative differences between the previous F-theory models associated with the magic square and the present case. We argue that the relevant half-hypermultiplets arise at the E6 points, where half-hypermultiplets 20 of SU(6) would have appeared in the split model. We also consider the problem on the non-local matter generation near the D6 point. After stating what the problem is, we explain why this is so by using the recent result that a split/non-split transition can be regarded as a conifold transition.

... Such structures lie inside of the realm of almost hypercomplex and almost quaternionic geometries, respectively. The aim of this note is to highlight them as the symplectic analogue of the well-known almost hypercomplex-Hermitian (hH) structures, almost quaternionic-Hermitian (qH) structures and their pseudo-Riemannian counterparts, which have been examined by many leaders in differential geometry, see for example [Aℓ68,Saℓ86,Sw91,AM96,Cor00,CabS10]. ...

We study $4n$-dimensional smooth manifolds admitting a $\mathsf{SO}^*(2n)$- or a $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure, where $\mathsf{SO}^*(2n)$ is the quaternionic real form of $\mathsf{SO}(2n, \mathbb{C})$. We show that such $G$-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better understood almost hypercomplex/quaternionic-Hermitian structures (hH/qH in short), where the underlying pseudo-Riemannian geometry is replaced by an almost symplectic geometry. We present several equivalent definitions of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon's $\mathsf{E}\mathsf{H}$-formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our $G$-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of homogeneous spaces $K/L$ with $K$ semisimple admitting an invariant torsion-free $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structure. This paper is the first in a series aiming at the description of the differential geometry of $\mathsf{SO}^*(2n)$- and $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures.

... They were classified in [Ale75] (see also [Cor96]). The Wolf spaces [Wol65] exhaust all compact, homogeneous quaternionic Kähler manifolds [Ale68], and it has been conjectured by LeBrun and Salamon [LS94] that this holds without assuming homogeneity, but only positive Einstein constant. ...

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.

... Examples for compact, quaternionic-Kähler manifolds, with positive scalar curvature are given by Wolf spaces. These are compact, homogeneous, quaternionic-Kähler manifolds classified by Wolf [12] and Alekseevskii [13]. The list includes quaternionic projective spaces ...

In this article, we establish a Hitchin-Kobayashi type correspondence for generalised Seiberg-Witten monopole equations on Kahler surfaces. We show that the "stability" criterion we obtain, for the existence of solutions, coincides with that of the usual Seiberg-Witten monopole equations. This enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors.

... For two distributions C ⊥ and Q ⊥ defined above, we may consider two natural invariant geometric properties under the shape operator A of M , that is, AC ⊥ ⊂ C ⊥ and AQ ⊥ ⊂ Q ⊥ . By using the result of Alekseevskii [1], Berndt and Suh [2, Theorem 1] have classified all real hypersurfaces with two natural invariant properties in G 2 (C m+2 ) as follows: ...

In this paper, we considered Ricci semi-symmetric real hypersurface in complex two-plane Grassmannians. Then we prove the non-existence of Ricci semi-symmetric Hopf hypersurfaces in complex two-plane Grassmannians by using the method of simultaneous diagonalization for pairwise commutative matrices.

... The corresponding geometrical feature is that the three dimensional distribution J(⊥M ) on M , which is obtained by applying the almost Hermitian structures in the quaternionic Kähler structure J of HP m to ⊥M , is invariant under A. In fact, every tube around a quaternionic submanifold of HP m has this geometrical feature. (Note that by a result of D.V. Alekseevskii [2] such a quaternionic submanifold is necessarily totally geodesic.) But the converse is not true. ...

In this paper the present author introduce some recent results about real hyper-surfaces in complex two-plane Grassmannians G2(C m+2) , which give an extended version of the talk delivered at the 8th International Workshop on Differential Geometry held at Kyungpook National University during the period from 12 December to 13 December 2003.

... By using such two geometric conditions and the results in Alekseevskii [1], Berndt and Suh [3] proved the following: M is an open part of a tube around a totally geodesic G 2 (C m+1 ) in G 2 (C m+2 ), or (B) m is even, say m = 2n, and M is an open part of a tube around a totally geodesic HP n in G 2 (C m+2 ). ...

In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians G(2)(Cm+2), m >= 3, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection (del) over cap ((k)).

... In Sections 3 and 4, the proofs of Theorem 2 and Corollary 2 will be given. Main references for Riemannian geometric structures of G 2 (C m+2 ), m ≥ 3 will be explained in detail (see [1], [2], [3], and [14]). ...

In this paper, we have introduced a new notion of generalized Tanaka-Webster
Reeb recurrent Ricci tensor in complex two-plane Grassmannians $G_2({\mathbb
C}^{m+2})$. Next, we give a non-existence property for real hypersurfaces $M$
in $G_2({\mathbb C}^{m+2})$ with such a condition.

... A quaternion Kaehler manifold M is a quaternion space form if its quaternionic sectional curvatures are constant. It is well known that a quaternion Kaehlerian manifold M is a quaternion space form M (c) if and only if its curvature tensor R is of the following form (see [1], ...

B. Y. Chen [Kodai Math. J. 4, 399–417 (1981; Zbl 0481.53046)] established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Recently Ximin Liu [Arch. Math. (Brno), 38, 297–305 (2002; Zbl 1090.53052)] obtained results on Ricci curvature of a totally real submanifold in a quaternion projective space extending the results of Chen. In this article, we wish to estimate the Ricci curvature of a quaternion CR-submanifold in a quaternion space form.

... Using two invariant conditions mentioned above and the result in Alekseevskii [1], Berndt and Suh [3] proved the following: ...

In this paper, we consider a new notion of Reeb parallel shape operator for real hypersurfaces \(M\) in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\). When \(M\) has Reeb parallel shape operator and non-vanishing geodesic Reeb flow, it becomes a real hypersurface of Type \((A)\) with exactly four distinct constant principal curvatures. Moreover, if \(M\) has vanishing geodesic Reeb flow and Reeb parallel shape operator, then \(M\) is model space of Type \((A)\) with the radius \(r = \frac{\pi }{4\sqrt{2}}\).

We prove that complete non-aspherical quaternionic Kähler manifolds with an end of finite volume exist in all dimensions $4m\ge 4$. These manifolds are not locally homogeneous and, in particular, not locally symmetric.

The internal consistency of string theory implies that the space time is eleven dimensional. In order to explain why we only observe four dimensions of space time we will make the assumption that the space time is a product of the space time that we observe with an internal compact space that is so small that we cannot observe it. A central problem of string theory is then to obtain effective theories that reproduce the standard model and general relativity. In my thesis I will focus on a technique to obtain such lower dimensional effective theories namely consistent truncations. The idea is to use the extended symmetries of string theory in order to select a finite set of modes involved in the effective theory inside the infinte set of reduced fields. In my thesis I will show how the formalism of generalised geometry, an extension of differential geometry that unifies space time coordinates transformation and gauge transformation of string theory potentials in generalised diffeomorphism, allow to obtain in a systematic way consistent truncations and thus classify effective theories that can be obtained from string theory. This method is general for any dimensions but in order to apply it to holographic duality between gauge theories and sting theory I will at some point specify to five dimensions reductions.

We study 4n-dimensional smooth manifolds admitting a SO*(2n)- or a SO*(2n)Sp(1)-structure, where SO*(2n) is the quaternionic real form of SO(2n, C). We show that such G-structures, called almost hypercomplex/quaternionic skew-Hermitian structures, form the symplectic analogue of the better known almost hypercomplex/quaternionic-Hermitian structures (hH/qH for short). We present several equivalent definitions of SO*(2n) and SO*(2n)Sp(1)-structures in terms of almost symplectic forms compatible with an almost hypercomplex/quaternionic structure, a quaternionic skew-Hermitian form, or a symmetric 4-tensor, the latter establishing the counterpart of the fundamental 4-form in almost hH/qH geometries. The intrinsic torsion of such structures is presented in terms of Salamon’s EH-formalism, and the algebraic types of the corresponding geometries are classified. We construct explicit adapted connections to our G-structures and specify certain normalization conditions, under which these connections become minimal. Finally, we present the classification of symmetric spaces K/L with K semisimple admitting an invariant torsion-free \SO*(2n)Sp(1)-structure. This paper is the first in a series aiming at the description of the differential geometry of \SO*(2n)- and SO*(2n)Sp(1)-structures.

We prove that complete non-locally symmetric quaternionic K\"ahler manifolds with an end of finite volume exist in all dimensions $4m\ge 4$.

We introduce the notion of generalized Killing Ricci tensor for real hypersurfaces in the complex hyperbolic quadric \({{Q^m}^*}={SO^0_{2,m}/SO_2 SO_m}\). We give a complete classification of real hypersurfaces in \({{Q^m}^*}={SO^0_{2,m}/SO_2 SO_m}\) with generalized Killing Ricci tensor.

We give a classification of real hypersurfaces in the complex hyperbolic quadric Qm∗=SO2,mo/SO2SOm that have constant mean curvature and harmonic curvature.

We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2 (ℂ m +2 ). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2 (ℂ m +2 ) and prove that a real hypersurface in G2 (ℂ m +2 ) with generalized Tanaka–Webster 𝔇 ⊥ -parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍ P ⁿ in G2 (ℂ m +2 ), where m = 2 n .

In this survey article, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 2. Second, by using the isometric Reeb flow, we give a complete classification for hypersurfaces M in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})=SU_{2+m}/S(U_{2}U_{m})\), complex hyperbolic two-plane Grassmannians \(G_{2}^{*}({\mathbb C}^{m+2})=SU_{2,m}/S(U_{2}U_{m})\), complex quadric \(Q^m={ SO}_{m+2}/SO_{m}SO_{2}\) and its dual \(Q^{m *}= SO_{m,2}^{o}/SO_{m}SO_{2}\). As a third, we introduce the classifications of contact hypersurfaces with constant mean curvature in the complex quadric \(Q^m\) and its noncompact dual \(Q^{m *}\) for \(m \ge 3\). Finally we want to mention some classifications of real hypersurfaces in the complex quadrics \(Q^m\) with Ricci parallel, harmonic curvature, parallel normal Jacobi, pseudo-Einstein, pseudo-anti commuting Ricci tensor and Ricci soliton etc.

We have considered a new notion of the shape operator A satisfies Killing tensor type for real hypersurfaces M in complex Grassmannians of rank two. With this notion we prove the non-existence of real hypersurfaces M in complex Grassmannians of rank two.

We prove the nonexistence of Hopf real hypersurfaces in complex two-plane Grassmannians such that the covariant derivatives with respect to Levi-Civita and kth generalized Tanaka–Webster connections in the direction of the Reeb vector field applied to the Riemannian curvature tensor coincide when the shape operator and the structure operator commute on the \(\mathcal Q\)-component of the Reeb vector field.

In this paper, we consider a new notion of generalized Tanaka–Webster D-parallel shape operator for a real hypersurface in a complex two-plane Grassmannian and prove a non-existence theorem of a real hypersurface.

We introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric . We give a complete classification, in terms of their -principal or their -isotropic unit normal vector fields, of real hypersurfaces in having harmonic curvature tensor.

A quaternionic Kahler manifold M is called positive if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kahler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic Kahler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic Kahler manifold with symmetry rank at least (m - 2) must be either isometric to HPm or Gr(2)(Cm+2), if m greater than or equal to 10.

We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2) which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove non-existence of real hypersurfaces in G2(ℂm+2) with generalized Tanaka-Webster parallel normal Jacobi operator. © 2015, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.

In this article, the author will study compact Kihler hypersurfaces $M$ in a complex Grassmann manifold $G_{r}(\mathbb{C}^{n})$ of $r$-planes, and give an upper bound for the first eigenvalue of the Laplacian (Theorem $\mathrm{A}$ ). In the case that $r=2$, $G_{2}(\mathbb{C}^{n})$ admits the quaternionic K\"ahler structure $\mathfrak{J}$ . When the tangent bundle $TM$ and the normal bundle $T^{[perp]}M$ of $M$ satisfy the property that $3T^{[perp]}M\subset TM,$ the author obtain sharper estimate (Theorem $\mathrm{B}$ ). It is an interesting problem that “What is $M$ satisfying $\mathfrak{J}T^{[perp]}M\subset TM$? ”. If $M$ is Einstein, without the assumption of homogeneity, we shall show that $M$ is congruent to a certain Kihler $\mathrm{C}$-space (Theorem $\mathrm{C}$ ). Theorems $\mathrm{A}$ , $\mathrm{B}$ and $\mathrm{C}$ are showed in the section 2, and are proved in later sections.

We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian G
2(ℂm+2). In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in G
2(ℂm+2) satisfying such conditions.

In this paper we consider the notion of ξ-invariant or D-fraktur sign⊥-invariant real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2) and prove that there do not exist such kinds of real hypersurfaces in G2(ℂm+2) with parallel second fundamental tensor on a distribution F-fraktur sign defined by F-fraktur sign = ξ ∪ D-fraktur sign⊥, where D-fraktur sign⊥ = Span {ξ1, ξ2, ξ3}.

It is known that submanifolds in Kaehler manifolds have many kinds of connections. Among them, we consider two connections, that is, Levi-Civita and Tanaka–Webster connections for real hypersurfaces in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\) . When they are equal to each other, we give some characterizations in \(G_2({\mathbb C}^{m+2})\) .

In this article, the author will study compact Kihler hypersurfaces $M$ in a complex Grassmann manifold $G_{r}(\mathbb{C}^{n})$ of $r$-planes, and give an upper bound for the first eigenvalue of the Laplacian (Theorem $\mathrm{A}$ ). In the case that $r=2$, $G_{2}(\mathbb{C}^{n})$ admits the quaternionic K\"ahler structure $\mathfrak{J}$ . When the tangent bundle $TM$ and the normal bundle $T^{[perp]}M$ of $M$ satisfy the property that $3T^{[perp]}M\subset TM,$ the author obtain sharper estimate (Theorem $\mathrm{B}$ ). It is an interesting problem that “What is $M$ satisfying $\mathfrak{J}T^{[perp]}M\subset TM$? ”. If $M$ is Einstein, without the assumption of homogeneity, we shall show that $M$ is congruent to a certain Kihler $\mathrm{C}$-space (Theorem $\mathrm{C}$ ). Theorems $\mathrm{A}$ , $\mathrm{B}$ and $\mathrm{C}$ are showed in the section 2, and are proved in later sections.

In this paper, we introduce a new notion of the generalized Tanaka-Webster invariant for a hypersurface M in G2(Cm+2), and give a non-existence theorem for Hopf hypersurfaces in G2(Cm+2) with generalized Tanaka-Webster invariant shape operator.

We consider a new notion of η-parallel shape operator in complex two-plane Grassmannians G 2 (ℂ m+2 ) and give a non-existence theorem for a Hopf hypersurface M in G 2 (ℂ m+2 ) with η-parallel shape operator.

In this article, we obtain sharp estimate of the Ricci curvature of quaternion slant, bi-slant and semi-slant submanifolds in a quaternion space form, in terms of the squared mean curvature.

A new notion of the generalized Tanaka-Webster D⊥-invariant for a hypersurface M in G2(ℂm+2) is introduced, and a classification of Hopf hyper-surfaces in G2(ℂm+2) with generalized Tanaka-Webster D⊥-invariant shape operator is given.

Let M be a differentiable manifold of class C
?. All tensor fields discussed below are assumed to be of class C
?. Let X be a vector field on M. If X vanishes at a point 0 ? M then X induces, in a natural way, an endomorphism a
X
of the tangent space V
o at 0. In fact if y ? V
0
and Y is any vector field whose value at 0 is y, then define a
x
y = [X, Y]0
. It is not hard to see that [X, Y]0
does not depend on Y so long as the value of Y at 0 is y.

In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian
manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem—Theorem 1 (the
proof of Theorem 1 became suggestive to us after we noted that the T
x
of [1] is just the a
x
of [6] when X is restricted to p
0, see [6], p. 539). In fact after introducing, below, the notion of one affine connection A on a manifold being rigid with respect to another affine connection B on M and making some observations concerning such a relationship, Theorem 1 is seen to be a reformulation of Theorem 2.

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the T x of [1] is just the a x of [6] when X is restricted to p 0 , see [6], p. 539).

Clifford-Wolf isometrien symmetrische Raume

- H. Freudental

Hotonomy and Lie algebras of motions in Riemannian manifolds T r a n s . A m e r

- B Kostant

B. Kostant, "Hotonomy and Lie algebras of motions in Riemannian manifolds," T r a n s. A m e r. Math. Soc., 8_~0, 528-542 (1955).

E s p a c e s homogeneous k~ihleriens

- A Lichnerowicz

A. Lichnerowicz, " E s p a c e s homogeneous k~ihleriens," Coll.Int. de Geom. Diff., Strassbourg, 171-184 (1953).

Riemann spaces with nonstandard holonomy groups

- D V Alekseevskii
- D. V. Alekseevskii

Espaces homogeneous kähleriens

- A Lichnerowicz