Quaternionic projective space pdf

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. Let m be a connected real hype rsurface in pm without boundary. The classical projective spaces real, complex, and quaternionic are studied in terms of their self maps, from a homotopy point of view. The complex projective line is also called the riemann sphere. Critical points of the willmore energy are called willmore surfaces in quaternionic projective space. Real hypersurfaces of quaternionic projective space. All these definitions extend naturally to the case where k is a division ring. We extend the discussion of projective group representations in quaternionic hilbert space which was given in our recent book.

Real hypersurfaces of quaternionic projective space satisfying u i r 0 u i r 0. Projective group representations in quaternionic hilbert space. The construction of a class of harmonic maps to quaternionic projective space james f. This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. In the category of topological groups, the group structure on s3 is unique up to isomorphism. Quaternionic projective space lecture 34 july 11, 2008 the threesphere s3 can be identi. Quaternionic projective space by tatsuyoshi hamada 0. On the symmetric squares of complex and quaternionic projective space yumi boote and nigel ray abstract. A characterization of quaternionic projective space by the. Self maps of iterated suspensions of these spaces are also considered. Jan 31, 2019 in this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space n. In this note we develop an alternative characterization of the quaternionic projective space using the conformalkilling equation.

Unless otherwise specified, all homology and cohomology is taken with integral coefficients, and for m an manifold, me hnm, dm. Chen department of mathematics, university of british columbia, vancouver, b. In this lecture, we will address the question of how canonical this structure is. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. On a quaternionic projective space yukio kametani and yasuyuki nagatomo received june 20, 1994 1.

Recall that a pform uon a riemannian manifold mm,g is killing, if. We use the cell decomposition of quaternionic projective space with one cell each in dimensions. Pdf willmore spheres in quaternionic projective space. Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of real dimension 4n. We prove the classi cation theorem for nonsuperminimal harmonic twotori in hp2 and hp3 theorems 5. On certain real hypersurfaces of quaternionic projective space. 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 mathematical journal, 2010. M i where each factor is a complex projective line and a complex hyperbolicline. A characterization of einstein real hypersurfaces in.

Quaternionic projective space and harmonic sequence let c2nbe a 2ndimensional complex number space with the standard hermitian. 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. Pdf the willmore energy for frenet curves in quaternionic projective space is the generalization of the willmore functional for immersions into the. Real hypersurfaces in quaternionic projective space.

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. Find materials for this course in the pages linked along the left. The goal in both cases is to classify, up to homology, all such maps. Detecting quaternionic maps between hyperkahler manifolds. 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. Willmore spheres in quaternionic projective space cern. The homology of quaternionic projective space is given as follows. 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. A characterization of pseudoeinstein real hypersurfaces in a quaternionic projective space hamada tatsuyoshi journal or publication title tsukuba journal of mathematics volume 20. If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The model 1 is asymptotically free 12 for arbitrary n 2.

Pdf we classify certain real hypersurfaces ot a quaternionic projective space satisfying the condition. The betti numbers of quaternionic projective space are thus for with and elsewhere. This fibration is easily obtaind when we identify s4 with hp1 and c2 with h. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. Finally we study the ricci tensor of a real hypersurface of quaternionic projective space and classify pseudoeinstein. 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. London mathematical society journals wiley online library. A characterization of pseudoeinstein real hypersurfaces in a. Let be a connected real dimensional submanifold of real codimension of a quaternionic kahler manifold with quaternionic kahler. Detecting quaternionic maps between hyperkahler manifolds jingyi chen 0 1 2 jiayu li 0 1 2 0 j. Homology of quaternionic projective space topospaces. 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 inverse image of every point of pv consist of two. A characterization of pseudoeinstein real hypersurfaces.

Using a baecklund transformation on willmore surfaces we generalize bryants result on willmore. It is a homogeneous space for a lie group action, in more than one way. 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 paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space n. Generalised connected sums of quaternionic manifolds. In the present paper, we treat the harmonic twotori in quaternionic projective space. In the compact case, this system is known to have the remarkable property that it admits a nonconstant.

In this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space hpn. On the ricci tensor of real hypersurfaces of quaternionic. Complex forms of quaternionic symmetric spaces 267 ii products m m1. A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space let s be the unit sphere in a normed vector space v, and consider the function. Twistorial maps between paraquaternionic projective spaces. In this paper the word manifold will always mean oriented compact c manifold. 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.

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 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. 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. In this paper we construct the generalizations of 1 using the quaternionic projective space qps hpn which arises naturally in four dimensions. 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. The quaternionic projective line is homeomorphic to the 4sphere. Our description involves generators and relations, and our methods entail ideas from. Principal angles and approximation for quaternionic projections loring, terry a. 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.