Our description involves generators and relations, and our methods entail ideas from. We study some conditions on the ricci tensor of real hypersurfaces of quaternionic projective space obtaining among other results an improvement of the main theorem in 9. London mathematical society journals wiley online library. A complete connected quaternionic kahler eightmanifold with t 0 is isometric to the quaternionic projective plane hp2, the complex grassmannian gr 2 c4, or the exceptional space g 2 so4.
The betti numbers of quaternionic projective space are thus for with and elsewhere. In the present paper, we treat the harmonic twotori in quaternionic projective space. Complex forms of quaternionic symmetric spaces 267 ii products m m1. On a real hypersurface of quatemionic projective space qpm we study the following condition. Real hypersurfaces of quaternionic projective space. M i where each factor is a complex projective line and a complex hyperbolicline. Generalised connected sums of quaternionic manifolds. Homology of quaternionic projective space topospaces.
Find materials for this course in the pages linked along the left. The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. In this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space hpn. Find, read and cite all the research you need on researchgate. In this paper the word manifold will always mean oriented compact c manifold. Pdf we classify certain real hypersurfaces ot a quaternionic projective space satisfying the condition. A characterization of pseudoeinstein real hypersurfaces in a quaternionic projective space hamada tatsuyoshi journal or publication title tsukuba journal of mathematics volume 20.
It is a homogeneous space for a lie group action, in more than one way. Principal angles and approximation for quaternionic projections loring, terry a. On the ricci tensor of real hypersurfaces of quaternionic. A characterization of pseudoeinstein real hypersurfaces. In this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space n. On certain real hypersurfaces of quaternionic projective space. The main purpose of thispaper is to provide a characterization of pseudoeinstein real hypersurface in hp by using an estimate of thelength of the ricci tensor s, which is a quaternionic version of a result of kimura and. Detecting quaternionic maps between hyperkahler manifolds.
The total space z is known as the twistor space of m and its complex and real structures together determine the quaternionic structure of the base. The homology of quaternionic projective space is given as follows. All these definitions extend naturally to the case where k is a division ring. Willmore spheres in quaternionic projective space cern. In the compact case, this system is known to have the remarkable property that it admits a nonconstant.
Schwarzenberger sc, hi has shown the fact that a kdimensional fvector bundle v over fpn for f r or c is stably equivalent to a whitney sum of k fline bundles if v is extendible, that is, if v is the restriction of a fvector bundle over fpm for any 111 n. Submanifolds of dimension in a quaternionic projective. Some characterizations of quaternionic space forms adachi, toshiaki and maeda, sadahiro, proceedings of the japan academy, series a, mathematical sciences, 2000 harmonic maps from the riemann sphere into the complex projective space and the harmonic sequences kawabe, hiroko, kodai. The purpose of this paper is to study ndimensional submanifolds of dimension in a quaternionic projective space and especially to determine such submanifolds under some curvature conditions. On real hypersurfaces of a quaternionic projective space, saitama math. Unless otherwise specified, all homology and cohomology is taken with integral coefficients, and for m an manifold, me hnm, dm. A characterization of quaternionic projective space by the.
We use the cell decomposition of quaternionic projective space with one cell each in dimensions. Quaternionic projective space by tatsuyoshi hamada 0. Using a baecklund transformation on willmore surfaces we generalize bryants result on willmore. Recall that a pform uon a riemannian manifold mm,g is killing, if. We prove the classi cation theorem for nonsuperminimal harmonic twotori in hp2 and hp3 theorems 5. On the symmetric square of quaternionic projective space march, 2016 the main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were rst proposed in the 1930s. Real, complex, quaternionic and octonionic projective spaces. This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. The complex projective line is also called the riemann sphere. Real hypersurfaces in quaternionic projective space.
Stably extendible vector bundles over the quaternionic. Glazebrook mathematics institute, university of warwick, coventry cv4 7al. Twistorial maps between paraquaternionic projective spaces. Quaternionic projective space lecture 34 july 11, 2008 the threesphere s3 can be identi. On a quaternionic projective space yukio kametani and yasuyuki nagatomo received june 20, 1994 1. The quaternionic projective line is homeomorphic to the 4sphere. If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. In this note we develop an alternative characterization of the quaternionic projective space using the conformalkilling equation. Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of real dimension 4n. The classical projective spaces real, complex, and quaternionic are studied in terms of their self maps, from a homotopy point of view. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. A characterization of einstein real hypersurfaces in.
Chen department of mathematics, university of british columbia, vancouver, b. Let be a connected real dimensional submanifold of real codimension of a quaternionic kahler manifold with quaternionic kahler. Then nis a totally geodesic submanifold of mwhich is isometric to a sphere if and only if nis a totally geodesic submanifold of a projective line of m. Self maps of iterated suspensions of these spaces are also considered.
In this paper we construct the generalizations of 1 using the quaternionic projective space qps hpn which arises naturally in four dimensions. Quaternionic projective space and harmonic sequence let c2nbe a 2ndimensional complex number space with the standard hermitian. Detecting quaternionic maps between hyperkahler manifolds jingyi chen 0 1 2 jiayu li 0 1 2 0 j. Finally we study the ricci tensor of a real hypersurface of quaternionic projective space and classify pseudoeinstein. Some characterizations of quaternionic space forms adachi, toshiaki and maeda, sadahiro, proceedings of the japan academy, series a, mathematical sciences, 2000 harmonic maps from the riemann sphere into the complex projective space and the harmonic sequences kawabe, hiroko, kodai mathematical journal, 2010. A characterization of pseudoeinstein real hypersurfaces in a. M, where each miis a the complex projective or hyperbolic plane with the quaternionic structure of complex scalar part, or b a product m i. According to our classification, more minimal constant curved twospheres in n are obtained than what ohnita conjectured in the paper homogeneous harmonic maps into complex projective spaces. The construction of a class of harmonic maps to quaternionic projective space james f. Jan 31, 2019 in this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space n. Let v denote the riemann connection of m and r the curvature tensor of the real hypersurface.
On connectedness of the space of harmonic 2spheres. Pdf qrsubmanifolds of p1 qrdimension in a quaternionic. Let hp be a quaternionic projective space, n 3, with metric g of constant quaternionic sectional curvature 4, and let m be a connected real hypersurface of hp. The inverse image of every point of pv consist of two. The goal in both cases is to classify, up to homology, all such maps. The model 1 is asymptotically free 12 for arbitrary n 2. We offer a solution for the complex and quaternionic projective spaces pn, by utilising their rich geometrical structure. We extend the discussion of projective group representations in quaternionic hilbert space that was given in our recent book. In this lecture, we will address the question of how canonical this structure is. We extend the discussion of projective group representations in quaternionic hilbert space which was given in our recent book. Let m be a connected real hype rsurface in pm without boundary. We consider frenet curves in quaternionic projective space and define the willmore energy of a frenet curve as a generalization of the willmore functional for immersions into the foursphere.
191 1038 566 641 77 326 1024 776 995 1444 611 1393 1045 1484 1160 365 1056 783 528 998 393 9 1084 55 305 1169 346 417 1421 550 956 125 671 1543 282 1249 37 903 1237 1176 16 904 70 497 151 1192 1468