 Embedding Condition 2: The notion of a tangent plane also requires that there is a one-to-one linear mapping between the line elements of \(\Sigma(M_{p})\) starting  from 0, with the line elements of \(M\) starting from \(p\).  This means that the number of dimension of the Klein space  \(\Sigma(M_{p})\) has the same number of dimension as the manifold \(M\).