Developed from the author's course at the U. of Notre Dame, this second-edition work (the first was published in 1996) covers roughly the same material as two of the five volumes of Spivak's A Comprehensive Introduction to Differential Geometry. Nicolaescu takes a more modern perspective on algebraic subjects, including sheaves and pre-sheaves, and a more abstract approah to connections in fiber bundles. Although he provides a generous number of examples of concrete calculations, students should have graduate level courses in analysis and topology under their belt. Exercises are included (no answers). Some typos and mistranslations occur in the text, but not in the math, and do not diminish the book's utility. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)