Synthetic Geometry of Manifolds
by Anders Kock
Publisher: University of Aarhus 2009
Number of pages: 314
This textbook can be used as a non-technical and geometric gateway to many aspects of differential geometry. The audience of the book is anybody with a reasonable mathematical maturity, who wants to learn some differential geometry.
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Differential geometry is deceptively simple. It is surprisingly easy to get the right answer with informal symbol manipulation. We use computer programs to communicate a precise understanding of the computations in differential geometry.
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We discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed.
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Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, and more.