On The Affine Geometric Properties Of Sl(m,r)/so(m)

As an important Lie group, SL(m,R)/SO(m) has many special properties and has been studied extensively. In the research of classifying the locally strongly convex affine hypersurfaccs with parallel cubic form in Rⁿ⁺¹, if n = 5, then SL(3,R)/SO(3) appears (sec[1]). That is, SL(3,R)/SO(3) with the canonical affine(metric) structure h has parallel cubic form.In this paper, we prove that for all m≥3, SL(m,R)/SO(m), as a hypcrsurfaccs in (?), with its canonical affine structure is a hyperbolic affine sphere with parallel cubic form. Moreover, SL(m,R)/SO(m) with its centro-affine metric is Einstein. It is worthwhile to point out that this gives the first known example of affine spheres whose centro-affine metric is Einstein but not having constant sectional curvature.