We study biharmonic homogeneous real hypersurfaces in complex projective space and quaternion projective space. We provide a classification of biharmonic homogeneous real hypersurfaces in quaternion projective space. We also classify pseudo-harmonic, subelliptic biharmonic, and Levi-harmonic homogeneous Hopf hypersurfaces in complex space forms.

Gyrations are operations on manifolds that arise in geometric topology, where a manifold M may exhibit distinct gyrations depending on the chosen twisting. For a given M, we ask a natural question: do all gyrations of M share the same homotopy type regardless of the twisting? A manifold with this property is said to have gyration stability. Inspired by recent work by Duan, which demonstrated that the quaternionic projective plane is not gyration stable with respect to diffeomorphism, we explore this question for projective planes in general. We obtain a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes up to homotopy.

We introduce quaternionic structures on abstract GKM graphs, as the combinatorial counterpart of almost quaternionic structures left invariant by a torus action of GKM type. In the GKM3 setting the 2-faces of the GKM graph can naturally be divided into quaternionic and complex 2-faces; it turns out that for GKM3 actions on positive quaternion-Kähler manifolds the quaternionic 2-faces are biangles or triangles, and the complex 2-faces triangles or quadrangles. We show purely combinatorially that any abstract GKM3 graph with quaternionic structure satisfying this restriction on the 2-faces of the GKM graph is that of a torus action on quaternionic projective space HPn or the Grassmannian Gr2(Cn) of complex 2-planes in Cn.

We prove that there do not exist odd-dimensional stable compact minimal immersions in the product of two complex projective spaces. We also prove that the only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold in the Riemannian product. These generalize similar results of the second-named author of immersions with low dimensions or codimensions to immersions with arbitrary dimensions. In addition, we prove that the only stable compact minimal immersions in the product of an octonionic projective plane with any other Riemannian manifold are the products of octonionic projective subspaces with compact stable minimal immersions of the second manifold in the Riemannian product.

Abstract In the previous paper (Skriganov, J. Complexity 56 (2020), 101428), Stolarsky's invariance principle, known in the literature for point distributions on Euclidean spheres, has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, a new pure analytic proof of the extended Stolarsky's invariance principle is given, relying on the theory of spherical functions on compact Riemannian symmetric manifolds of rank one.

We investigate the smooth convergence of the mean curvature flow of hypersurfaces in the quaternionic projective spaces.We prove that if the initial hypersurface satisfies a new nonlinear curvature pinching condition, then the mean curvature flow converges smoothly to a round point in finite time.Our result improves a smooth convergence theorem due to Pipoli and Sinestrari (2017).

On the stability of the quaternion projective space

Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces ℍP n ,n≥3, and on the Cayley projective plane 𝕆P 2 .

In this article we study the concept of quaternionic bisectional curvature introduced by Chow and Yang in (J Differ Geom 29(2):361–372, 1989) for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space HPn. We also show that all symmetric quaternion-Kähler manifolds different from HPn admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

This paper is a survey on the classification of minimal two-spheres with constant curvature in somesymmetric spaces, including the classification of homogeneous and homogeneous minimal two-spheres in the complex Grassmann manifoldG(2,n), the complex hyperquadric Q_n and the quaternionic projective space mathbbHP^n. We also introduce some problems.

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models.

constructed three 15-vertex combinatorial 8manifolds 'like the quaternionic projective plane' with symmetry groups A 5 , A 4 , and S 3 , respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to HP 2 . Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to S 8 . In the present paper we construct a lot of new 15-vertex triangulations of HP 2 . A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group A 5 . Namely, we find 19 triangulations with symmetry group C 7 , one triangulation with symmetry group C 6 × C 2 , 14 triangulations with symmetry group C 6 , 26 triangulations with symmetry group C 5 , one new triangulation with symmetry group A 4 , and 11 new triangulations with symmetry group S 3 . Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of HP 2 with 15 vertices and symmetry group of order at least 4: the three Brehm-Kühnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups C 3 and C 2 , as well as the trivial symmetry group.

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

Abstract Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{H}\mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

В 1987 г. У. Брем и В. Кюнель показали, что всякая триангуляция $d$-мерного многообразия (без края), не гомеоморфного сфере, имеет не меньше $3d/2+3$ вершин. Более того, триангуляции ровно с $3d/2+3$ вершинами могут существовать только для "многообразий, похожих на проективные плоскости", которые бывают только в размерностях $2$, $4$, $8$ и $16$. Имеются $6$-вершинная триангуляция вещественной проективной плоскости $\mathbb{RP}^2$, $9$-вершинная триангуляция комплексной проективной плоскости $\mathbb{CP}^2$ и $15$-вершинные триангуляции кватернионной проективной плоскости $\mathbb{HP}^2$. Недавно автор построил первые примеры $27$-вершинных триангуляций многообразий, похожих на октавную проективную плоскость $\mathbb{OP}^2$. Четыре наиболее симметричные из них имеют группу симметрий $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ порядка $351$. Эти триангуляции были найдены при помощи компьютерной программы после того, как была угадана их группа симметрий. Тем не менее оставалось совершенно непонятным, почему именно эта группа реализуется как группа симметрий и существуют ли $27$-вершинные триангуляции многообразий, похожих на $\mathbb{OP}^2$, с другими (возможно, большими) группами симметрий. В настоящей работе даются сильные ограничения на группы симметрий таких $27$-вершинных триангуляций. А именно, приводится список из $26$ подгрупп симметрической группы $\mathrm{S}_{27}$, содержащий все возможные группы симметрий $27$-вершинных триангуляций многообразий, похожих на октавную проективную плоскость. (Нам не известно, все ли эти подгруппы реализуются как группы симметрий.) Группа $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ является самой большой в этом списке, причем порядки всех остальных групп не превосходят $52$. Ключевую роль в нашем подходе играет использование результатов П. Смита и Г. Бредона о топологии множеств неподвижных точек конечных групп преобразований. Библиография: 36 названий.

In 1987 Brehm and Kühnel showed that any triangulation of a $d$-manifold (without boundary) that is not homeomorphic to a sphere has at least $3d/2+3$ vertices. Moreover, triangulations with exactly $3d/2+3$ vertices can exist only for ‘manifolds like projective planes’, which can have dimension $2$, $4$, $8$ or $16$ only. There is a $6$-vertex triangulation of the real projective plane $\mathbb{RP}^2$, a $9$-vertex triangulation of the complex projective plane $\mathbb{CP}^2$ and $15$-vertex triangulations of the quaternionic projective plane $\mathbb{HP}^2$. Recently the author constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane $\mathbb{OP}^2$. The four most symmetric of them have the symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$. These triangulations were constructed using specially designed software after the symmetry group had been guessed. However, it remained unclear why exactly this group is realized as a symmetry group and whether $27$-vertex triangulations of manifolds like $\mathbb{OP}^2$ with other (possibly larger) symmetry groups exist. In this paper we find strong restrictions on the symmetry groups of such $27$-vertex triangulations. Namely, we present a list of $26$ subgroups of $\mathrm{S}_{27}$ containing all possible symmetry groups of $27$-vertex triangulations of manifolds like the octonionic projective plane. (We do not know whether all these subgroups can be realized as symmetry groups.) The group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ is the largest group in this list, and the orders of all other groups do not exceed $52$. A key role in our approach is played by the use of results of Smith and Bredon on the topology of fixed-point sets of finite transformation groups. Bibliography: 36 titles.

The study of Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian [Formula: see text] is first investigated. It is proved that there exists a shrinking Ricci-Bourguignon soliton on a Hopf real hypersurface [Formula: see text] in [Formula: see text] by using pseudo-anticommuting Ricci tensor. Moreover, we have proved that there does not exist a nontrivial gradient Ricci-Bourguignon soliton ([Formula: see text]) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian [Formula: see text]. Among the class of contact hypersurface in [Formula: see text], we also prove that there does not exist a nontrivial gradient Ricci-Bourguignon in [Formula: see text] over the totally geodesic and totally real quaternionic projective space [Formula: see text] in [Formula: see text], [Formula: see text].

We construct the four-derivative supersymmetric extension of (1, 0), 6D supergravity coupled to Yang-Mills and hypermultiplets. The hypermultiplet scalars are taken to parametrize the quaternionic projective space Hp(n) = Sp(n, 1)/Sp(n) × Sp(1)R. The hyperscalar kinetic term is not deformed, and the quaternionic Kähler structure and symmetries of Hp(n) are preserved. The result is a three parameter Lagrangian supersymmetric up to first order in these parameters. Considering the case of Hp(1) we compare our result with that obtained from the compactification of 10D heterotic supergravity on four-torus, consistently truncated to N = (1, 0), in which the hyperscalars parametrize SO(1, 4)/SO(4). We find that depending on how the Sp(1) is embedded in the SO(4), the results agree for a specific value of the parameter that governs the higher derivative hypermultiplet couplings.

Suppose that M is a closed isotropic Riemannian manifold and that R_1,\dots ,R_m generate the isometry group of M . Let f_1,\dots ,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from S^n to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.