On supersymmetric solutions in D = 11 supergravity with Ricci-flat internal spaces

An introduction to supersymmetric (SUSY) solutions defined on the product of Ricci-flat spaces in D= 11 supergravity is presented. The paper contains some background information: (i) decomposition relations for SUSY equations and (ii) 2^{-k}-splitting theorem that explains the appearance of N = 2^{-k} fractional supersymmetries. Examples of M2 and M5 branes on the product of two Ricci-flat spaces are considered and formulae for (fractional) numbers of unbroken SUSY are obtained.