  For a Riemannian type of metric structure the congruent linear maps  of the tangent space T\((M_{p})\) onto itself form  a group \(G_{p}\) which is isomorphic to the  orthogonal group \(O(n)\). The Pythagorean-Riemannian  metric at \(p\) is therefore determined through the concrete  realization of the orthogonal group at \(p\) which leaves the fundamental quadratic differential form at \(p\) invariant.  Thus the Pythagorean-Riemannian type of metric is  characterized by the abstract microsymmetry group \(O(n)\). For a metric which is not of the Pythagorean-Riemannian metric type,  the abstract microsymmetry group \(G_{p}\) will be  different from \(O(n)\) and will be some other subgroup of  \(GL(n)\). At each point of the manifold the microsymmetry  group will be a concrete realization of this subgroup of  \(GL(n)\). Weyl now states what he calls 