By Anatole Katok, Vaughn Climenhaga

Surfaces are one of the commonest and simply visualized mathematical gadgets, and their research brings into concentration primary rules, suggestions, and techniques from geometry, topology, advanced research, Morse thought, and staff conception. whilst, a lot of these notions look in a technically easier and extra picture shape than of their basic ``natural'' settings. the 1st, basically expository, bankruptcy introduces a few of the central actors--the around sphere, flat torus, Mobius strip, Klein bottle, elliptic airplane, etc.--as good as quite a few tools of describing surfaces, starting with the normal illustration through equations in three-d house, continuing to parametric illustration, and in addition introducing the fewer intuitive, yet valuable for our reasons, illustration as issue areas. It concludes with a initial dialogue of the metric geometry of surfaces, and the linked isometry teams. next chapters introduce primary mathematical structures--topological, combinatorial (piecewise linear), gentle, Riemannian (metric), and complex--in the categorical context of surfaces. the focus of the e-book is the Euler attribute, which seems to be in lots of diversified guises and ties jointly strategies from combinatorics, algebraic topology, Morse conception, traditional differential equations, and Riemannian geometry. The repeated visual appeal of the Euler attribute offers either a unifying subject and a strong representation of the concept of an invariant in all these theories. The assumed heritage is the normal calculus series, a few linear algebra, and rudiments of ODE and actual research. All notions are brought and mentioned, and nearly all effects proved, in line with this historical past. This ebook is as a result of the the MASS path in geometry within the fall semester of 2007.