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Smooth manifolds - University of Arizona Math
Smooth manifolds - University of Arizona Math

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Lecture 2. Smooth functions and maps

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MATH0055 2. 1. (a) What is a topological space? (b) What is the

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IN-CLASS PROBLEM SET (1) Find a continuous surjection f : R → {a

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Topology Ph.D. Qualifying Exam Alessandro Arsie, Gerard Thompson and Mao-Pei Tsui

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PDF file without embedded fonts

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Topology Ph.D. Qualifying Exam Gerard Thompson Mao-Pei Tsui April 14, 2007

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RIEMANN SURFACES 2. Week 2. Basic definitions 2.1. Smooth

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Lecture Notes 2

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Math 4853 homework 29. (3/12) Let X be a topological space

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Chapter 1: Some Basics in Topology

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Lecture 4(30.01.09) Universal Bundles

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Homework M472 Fall 2014

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GEOMETRY OF SURFACES b3 course 2004 Nigel Hitchin

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Orientability



In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a ""clockwise"" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a ""clockwise"" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as 20px in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image 20px.The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
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