The Gauss-Bonnet-Chern type formulas and their application in general relativity

In the article an overview of Gauss-Bonnet-Zhen type formulas, some of which were included in the 2009 thr author's review, some of which were missing from the mentioned review, and some were obtained later is given. Special attention is paid to the application of these formulas in the general theory of relativity. The positive energy required to create a 4-dimensional wormhole through which a warp ship flies has been established. The conclusions of the formula in supersymmetric Euclidean field theory and from the Atiyah- Singer index theorem are given.