Non-Existence Of Quadratic Harmonic Maps Of S~5 Into S~5

At the end of last century,the harmonic mapping between two manifolds is a hot research subject.Harmonic mapping is the generalization of geodesic and minimal manifold.In particular,when the source manifold and the image manifold are both spheres,the study about them is also certainly interesting.In this paper,we settle the case of non-existence of full quadratic harmonic maps from S5 to S5.Firstly,assume that there exists full quadratic harmonic maps from S5 to S5.Then,we use this assumption and mainly by performing othogonal transformantions and comparing the coefficients to deduce a contradiction.There are a lot of results about ?2-eigenmap between high dimensional spheres.However,there are still many problems in ?2-eigenmap even between lower dimensional spheres.H.X.He etc.proved that let f:S2n-4 ? Sn be a non-constant?2-eigenmap,then n is equal to 4 or 8.And that there is a non-constant?2-eigenmap f:S2n-5?Sn(n?6)if and only if n=6,8,9 or 10.Nevertheless,the existence of f:S5?S5 was not included in their paper.In this paper,I will settle the case of non-existence of full quadratic harmonic maps from S5 to S5.Inspired by Professor Wu's article,by using similar methods,I give a proof of our result.Firstly,assume that there exists full quadratic harmonic maps from S5 to S5.Then,extend the ?2-eigenmap between Eucilidean spheres into a corresponding quadratic harmonic map F:R6 ?R6.To solve problem,by dimension reduction,we obtain a mapping between 5 dimensional spaces from a map between 6 dimensional spaces.Furthermore,we express the map as the sum of three vector functions.Based on the derived vector functions and comparing the coefficients,we obtain a lot of identities.Finally,with a detailed discussion of these identities,we arrived at contradictions.Consequently,we have the conclusion that there is no full ?2-eigenmaps f:S5?S5.