Some Particular Surfaces In A Quaternionic Projective Space

Particular surfaces contain the surfaces whose Gauss curvature is constant and average curvature is zero. The paper study mainly the special surfaces in a quternionic projective space-minimal surfaces.Minimal surfaces is the surface whose average curvature is zero. It is derived by Plateau question first. It is an important problem which has long history in maths. It is a curvature which be studied mostly in Differential Geometry, it has deeply relation with compound analytic function and partial differential equation. It has a osculation relation with many mathematics embranchment and has important application in the field such as architecture, aviation, seteamboat manufacture, biology and computer graphics. Because of it wide application,when this question was put foward in 1820s it was attracting many scholar. The representation people has Jesse, Douglas, Morse, Couant and so on. They studied deeply in the questions whether minimal surfaces are exist?On which conditions it exist?How many minimal surfaces there are? This paper using the way which prehuman are used and the way which are at large used in surfaces study to make a study in the minimal surfaces in a quternionic projective space.When study the minimal surfaces in a quternionic projective space, first we should know well of the quaternionic projective space. The quaternionic projective space is a quaternionic Kahler manifold which have a constant quaternionic sectional curvature 4. Owing to the excommute of the quaternion, it is difficult to study the quaternionic projective space. So we make as forefathers, explain the quaternion field H to a two dimension complex vector space C², and bring a real linear operatorj : C²â†’C² satisfied j² = -1. Thus, we can introduce measure in quaternionic projective space HPⁿ, make it to a quaternionic Kahler manifold which have a constantquaternionic sectional curvature 4.About the theme of minimal surfaces, Xingxiao Li, Ximing Liu and other man have engaged deeply in the minimal immersion in quaternionic projective space of surfaces, but as to the surfaces in HPⁿ, results are not so many. Salamon studied the inclusive minimal surfaces in quaternionic Kahler manifold, proved this kind of surfaces can be lifted to the minimal surfaces in (?). Glazebrook studed isotropic harmonic mapping from surfaces to HPⁿ, proved the set of linear complete isotropic harmonic mappingφ: M→HPⁿ and the set of one kind of holomorphic subbundle in M×C² are monogamy. Zandi defined smoothing quaternionic Darboux frame on the minimal surfaces in quaternionic Kahler manifold, the frame is continuous on degenerat point, then defined a 6 degree (6,0)form differential expression, and proved the differential expression is holomorphic in the situation of HPⁿThis paper is divided into 4 parts:Chaper1 is introduction, introduce history origin of minimal surfaces and the main contents of the paper.Chapter2 is quaternionic projective space, introduce quaternionic linear space, then give a definition of quaternionic projective space and the connection on quaternionicprojective space.Chapter3 use the local lifting from x : M→HPⁿ to S⁴ⁿ⁺³ and method of moving frames study the surfaces in quaternionic projective space, give the mean curvature in quaternionic projective space and the necessary and sufficient condition of x : M→HPⁿ is minimal surfaces.Chapter4 give the definition of constant quaternionic angle in quaternionic projectivespace first,then give the theorem: assume x : M→HPⁿ is minimal surfaces of constant quaternionic angle 0 <θ<Ï€/2, then the M whose Gaussian curvature≤1, and equidistance to the minimal surfaces of constant quaternionic angle in HP² when k = 1, then prove it.