By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

**Read or Download A Primer on Riemann Surfaces PDF**

**Best geometry books**

**Fractals Everywhere: The First Course in Deterministic Fractal Geometry**

This version additionally gains extra difficulties and instruments emphasizing fractal functions, in addition to a brand new solution key to the textual content routines.

**Dynamical Systems X: General Theory of Vortices**

The English train mechanics as an experimental technological know-how, whereas at the Continent, it has regularly been thought of a extra deductive and a priori technological know-how. surely, the English are correct. * H. Poincare, technology and speculation Descartes, Leibnitz, and Newton As is widely known, the elemental ideas of dynamics have been said by way of New ton in his well-known paintings Philosophiae Naturalis Principia Mathematica, whose booklet in 1687 was once paid for via his buddy, the astronomer Halley.

The aim of this publication is to supply an creation to the speculation of jet bundles for mathematicians and physicists who desire to examine differential equations, rather these linked to the calculus of diversifications, in a latest geometric approach. one of many topics of the ebook is that first-order jets will be regarded as the ordinary generalisation of vector fields for learning variational difficulties in box idea, and such a lot of of the structures are brought within the context of first- or second-order jets, prior to being defined of their complete generality.

- Lectures on Algebraic Geometry II: Basic Concepts, Coherent Cohomology, Curves and their Jacobians
- Fractals, chaos, power laws: minutes from an infinite paradise
- The Spectrum of Hyperbolic Surfaces
- Lie Groupoids and Lie Algebroids in Differential Geometry

**Additional info for A Primer on Riemann Surfaces**

**Example text**

5. For any surface S, the following conditions 37 are equivalent: (1) the topology on S has a countable base? (2) S has a countable open cover of parametric discs? 'K 2' •• • of s with K. c K 0 c ... L , UK n Z = S. Proof. }. For each j, choose, wherever possible, a single parametric disc Q . with B . c q .. } is a countable collection of open 3 3 3 3 parametric discs. ) and there is some B . , B . c q . Thus x 3 3 3 3 and so uQ^ = S which proves (2). As each parametric disc is Qj homeomorphic to a disc in I, it has a countable base.

Is any set of the form and Y A x b where respectively. Clearly Z itself A and B are is an open rectangle: also, a finite intersection of open rectangles is an open rectangle. 1 that the class of open rectangles is a base for some topology on Z: we call this the product topology on Z. There are natural coordinate maps P 1 : (x,y) of Z onto X and Y h- x It follows that if h- y respectively and these are continuous because (p )_ 1 (A) = A x Y Plf , P 2^« P 2 : (x,y) , , f : W -*■ Z (p2)_ 1 (B) = X * B .

Unfortunately, as a general procedure this does not always yield the correct quotient space (it does here but in general, we need extra hypotheses) so we must seek a more explicit procedure Each point of the plane can be expressed uniquely in the form aA + by let S xs where S xs a and b are real. Let S be the unit circle in be the corresponding product space. There is map of (D and onto given by . 2). 2. 1), we see that G/G is homeomorphic to S*S. As a torus is obtained by rotating Sxs a circle around an axis, it can be parametrised by so G/G is topologically a torus.