I'll mention WL Burke's "Applied Differential Geometry." It's written for physicists, it will not be to the liking of the majority of mathematicians, but it changed this engineer's view of geometric methods forever.

The book changed the direction of my research because it presented a point of view that is not readily accessible if you follow the control and optimization literature. Becoming familiar with the differential geometry literature is an investment that a controls person is unlikely to make without a general idea of where the complete set of tools leads to. In this sense, the mathematics literature can present an obstacle. Burke's exposition is intuitive, though quite informal, and led me to read Spivak, Milnor, and other books, some mentioned here, which I would not have read if I had started with the math literature.