The paper has a very good historical and astrophysical introduction. Meanwhile, a reference to works by D. Richstone, A. Dressler and J. Kormendy [see, e.g., Bull. Am. Astron. Soc. 19, p. 713; Astrophys. J. Part 1 335, 532–541 (1988)], which put the beginning for the searching of supermassive black holes in the nuclei of galaxies will only add to the quality of the paper under review. The authors cited above, after analysing the luminosity curves near the cores of some of the galaxies with a very good angular resolution, found that some of the galaxies could have black holes in their nuclei. Subsequently the number of galaxies with black holes in their nuclei increased. Then, namely these authors could apply for discovery of supermassive black holes, a dream of the humans since the epoch-making work by Karl Schwarzschild (1916). Those papers need a special attention in view of their importance for the discoveries of supermassive black holes. Bang-Yen Chen is generalizing the studies of marginally trapped surfaces (horizons of events) for Kaluza-Klein theories, which played a very important role in the 4-dimensional black holes studies. The concept, which was introduced by Roger Penrose (1965) was a cornerstone for the achievement of singularity theorems, the analysis of the gravitational collapse, which was given by J. A. Wheeler, M. Masami, R. Ruffini and others.
In this article the author is surveying recent classification results of marginally trapped surfaces from a differential geometric viewpoint. The author is giving a brief introduction to a closely related subject, namely, the Kaluza-Klein theory. In the final part of the article several different recent approaches to the Kaluza-Klein theory without using compactification of the extra dimensions is given.