Linear isotropy group of an affine symmetric space

TLDR: In this article, a necessary and sufficient condition that a simply connected affine symmetric space M such that K coincides with the linear isotropy group dHp at some point p in M is given.

Abstract: Let K be a subgroup of the general linear group GL(n). The author found a necessary and sufficient condition that there exist an «-dimensional simply connected affine symmetric space M such that K coincides with the linear isotropy group of all affine automorphisms of M at some point in M. Let M he an w-dimensional manifold with affine connection, A(M) the group of all affine automorphisms of M, HB the subgroup of A(M) consisting of all elements of A(M) which fix a point p in M, and dHv the linear isotropy group determined by Hp. Let V be an «-dimensional vector space, GL(n) the general linear group of V, and K a subgroup of GL(n). We shall find a necessary and sufficient condition that there exists a simply connected affine symmetric space M such that K coincides with the linear isotropy group dHp at some point p in M. We discussed similar problems for a Riemannian symmetric space [6]. First of all we shall prove the following: Lemma. Let T be a tensor in V®V*®V*®V* which satisfies the following conditions. (1) T,jkl = — T.jtk, (2) r.m + r.klj + r.lik = o, / 3\ -TU 'T'A 'T'A fi T-A nri T-A T-l _ A \D) i -hmn1 ikl l -jmn1 -hkl * -kmn1 -jhl L -tinn1 -ikh — u> where T.'m are the components of T. Then there is an affine symmetric space whose curvature tensor at some point of it coincides with T. Proof. We integrate the following differential equations. dcofdt = da' + akô4, dô^/dt = T.kjla¡of, with initial conditions (w')(=0=0, ((74)i=0=0. The solutions ¿ô*, œ'k are linear forms in da1, • • ■ , da" whose coefficients are integral functions of t, a1, • • ■ , an. If we set r=l and replace a1 by x\ we have forms tü'(jc, dx), w)(x, dx). Since the determinant of the Received by the editors June 16, 1971. AMS 1970 subject classifications. Primarv 53C35.