By Anatole Katok, Vaughn Climenhaga

Surfaces are one of the commonest and simply visualized mathematical gadgets, and their examine brings into concentration basic rules, techniques, and techniques from geometry, topology, advanced research, Morse conception, and workforce thought. even as, lots of these notions look in a technically less complicated and extra picture shape than of their basic ``natural'' settings. the 1st, essentially expository, bankruptcy introduces a few of the valuable actors--the around sphere, flat torus, Mobius strip, Klein bottle, elliptic aircraft, etc.--as good as numerous equipment of describing surfaces, starting with the conventional illustration via equations in third-dimensional area, continuing to parametric illustration, and likewise introducing the fewer intuitive, yet primary 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), soft, Riemannian (metric), and complex--in the categorical context of surfaces. the point of interest of the publication is the Euler attribute, which seems to be in lots of diverse guises and ties jointly techniques from combinatorics, algebraic topology, Morse concept, traditional differential equations, and Riemannian geometry. The repeated visual appeal of the Euler attribute presents either a unifying topic and a strong representation of the idea of an invariant in all these theories. The assumed history is the traditional 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 heritage. This booklet is as a result the MASS direction in geometry within the fall semester of 2007.

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Then the quotient space of X by the action of τ is the cylinder; that is, we identify each point with all of its images under iterates of τ . Then a square root of τ is given by σ : (x, y) → (x + 1, −y), and the quotient space of X by the action of σ is the M¨obius strip. The covering map arises naturally as the canonical projection f : X/σ 2 → X/σ {(x + 2n, y) : n ∈ Z} → {x + n, (−1)n y : n ∈ Z} A similar argument, whose details are left to the reader, shows that the torus is a double cover for the Klein bottle.

A0 . . a0 . , as shown in figure 33. Following the same procedure as in the proof of the 62 2. COMBINATORIAL STRUCTURE AND TOPOLOGICAL CLASSIFICATION theorem, we may remove a M¨obius cap from the surface and replace it with a disc to obtain a planar model, with fewer edges, of a surface S ′ . Figure 33. The configuration a0 a0 If S ′ is orientable, we apply the theorem and have that our original surface S can be represented by the identifications −1 −1 −1 a0 a0 a1 b1 a−1 1 b1 . . am bm am bm If S ′ is not orientable, we use the same argument to remove another M¨obius cap, and continue until we obtain either an orientable surface or the projective plane aa.

Then a square root of τ is given by σ : (x, y) → (x + 1, −y), and the quotient space of X by the action of σ is the M¨obius strip. The covering map arises naturally as the canonical projection f : X/σ 2 → X/σ {(x + 2n, y) : n ∈ Z} → {x + n, (−1)n y : n ∈ Z} A similar argument, whose details are left to the reader, shows that the torus is a double cover for the Klein bottle. We repeat our observation from a previous lecture that the Euler characteristics of S and S˜ are related; in particular, if S˜ is an n-fold cover of S, we have ˜ = nχ(S) χ(S) These examples point us towards a general result concerning non-orientable surfaces.