Lectures on Surfaces (Student Mathematical Library, Volume 46)

By Anatole Katok, Vaughn Climenhaga

Surfaces are one of the commonest and simply visualized mathematical gadgets, and their examine brings into concentration primary principles, strategies, and strategies from geometry, topology, advanced research, Morse concept, and crew thought. even as, a lot of these notions look in a technically easier and extra photo shape than of their common ``natural'' settings. the 1st, basically expository, bankruptcy introduces some of the significant actors--the around sphere, flat torus, Mobius strip, Klein bottle, elliptic aircraft, etc.--as good as a variety of tools of describing surfaces, starting with the conventional illustration by means of equations in 3-dimensional area, 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 basic mathematical structures--topological, combinatorial (piecewise linear), soft, Riemannian (metric), and complex--in the explicit context of surfaces. the point of interest of the ebook is the Euler attribute, which seems in lots of various guises and ties jointly techniques from combinatorics, algebraic topology, Morse conception, usual differential equations, and Riemannian geometry. The repeated visual appeal of the Euler attribute presents either a unifying subject matter and a strong representation of the proposal of an invariant in all these theories. The assumed history is the normal calculus series, a few linear algebra, and rudiments of ODE and genuine research. All notions are brought and mentioned, and nearly all effects proved, in line with this heritage. This booklet is due to the the MASS path in geometry within the fall semester of 2007.

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The matter with this photo is this aircraft could intersect the outside at different issues in addition, and should definitely intersect different tangent 107 108 four. RIEMANNIAN METRICS planes, even supposing we wish to give some thought to the tangent package as being the disjoint union of the tangent areas. this is often more straightforward to visualize if we think about a one-dimensional manifold, the circle. Then the tangent area at each one aspect is just a line, and if we connect disjoint strains to every aspect on a circle, we receive a cylinder, a noncompact two-dimensional manifold, because the tangent package deal of S 1 .

LECTURE four 15 these triples (x, y, z) for which the triangle inequality turns into degenerate, that's, for which d(x, z) = d(x, y) + d(y, z). for instance, if our area X is simply the Euclidean airplane R2 with distance functionality given by way of Pythagoras’ formulation, d((x1 , x2 ), (y1 , y2 )) = (y1 − x1 )2 + (y2 − x2 )2 then the triangle inequality is a final result of the Cauchy-Schwarz inequality, and we've got equality within the one iff we've got equality within the different; this happens iff y lies within the line phase [x, z], in order that the 3 issues x, y, z are in reality collinear.

To this finish, examine a floor with a triangulation T , or extra ordinarily, any simplicial advanced. we are going to outline a sequence complicated C(T ), research the geometric interpretation of the areas Ck (T ) and the boundary operators ∂k , and discover a notable courting among the Euler attribute χ(T ) and the Betti numbers βk . whereas the algebraic definition of C assigned no specific interpretation to the indices ok, for our reasons the following they're to be regarded as indicating the size of the gadgets from which Ck , Hk , βk , and so on.

We will preserve this symmetry through embedding in R4 , the so-called flat torus. Parametrically, this can be given through x y z w = = = = r cos t r sin t r cos s r sin s the place s, t ∈ [0, 2π]. we will additionally receive the torus as an element house, utilizing an identical process as within the definition of the projective airplane or Klein bottle. starting with a rectangle, we determine contrary facets (with no reversal of direction); alternately, we will be able to examine the kinfolk of isometries of R2 given via Tm,n : (x, y) → (x + m, y + n), the place m, n ∈ Z, and mod out through orbits.

We will be able to make this unique as follows: Definition 2. A triangulation of a floor S is a suite T of 2simplices, T = {σi2 }ni=1 , such that the next carry: (1) S = ni=1 σi2 1, (2) for each i = j, the intersection σi2 ∩ σj2 is both a 1-simplex σij zero , or the empty set ∅. a 0-simplex σij 2. 1. LECTURE eight: FRIDAY, SEPT. 14 33 (3) each 1-simplex is within the boundary of precisely of the σi2 ; that's, 1 = σ 1 iff (i, j) = (k, ℓ). σij okℓ (4) each 0-simplex is within the boundary of numerous σi2 that may be prepared in a cyclic order; that's, given σ zero , the set of σi2 which include σ zero will be installed an inventory σi21 , .

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