MAXIMAL TORI OF EXTRINSIC SYMMETRIC SPACES AND MERIDIANS (vol 12, pg 564, 2022)

Improving a theorem in [1] we observe that a maximal torus of an extrinsic symmetric space in a euclidean space V is itself extrinsic symmetric in some affine subspace of V. A compact extrinsic symmetric space is a submanifold X C Sp-1 C Rp = V such that for any point x E X the reflection sx along the normal space N = NxX keeps X invariant. Every compact symmetric space X contains a maximal torus T which is unique up to congruence. If X = Sn C Rn+1, the maximal torus is a great circle C = X n R2 which is reflective, hence extrinsic symmetric, see [1, Theorem 4]. But for most extrinsic symmetric spaces, the maximal torus is not reflective. However, as we will show, it is an "iterated" reflective subspace, and in particular