chap05 - manifolds

chap05 - manifolds - Manifolds 1 Chapter 5 Manifolds We are...

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Unformatted text preview: Manifolds 1 Chapter 5 Manifolds We are now going to begin our study of calculus on curved spaces. Everything we have done up to this point has been concerned with what one might call the flat Euclidean spaces R n . The objects that we shall now be investigating are called manifolds . Each of them will have a certain dimension m . This is a positive integer that tells how many independent “coordinates” are needed to describe the manifold, at least locally. For instance, the surface of the earth is frequently modeled as a sphere, a 2-dimensional manifold, with points located in terms of the two quantities latitude and longitude. (This description clearly holds only locally — for instance, the north pole is described in terms of latitude = 90 ◦ and longitude is undefined there. Further, longitude ranges between- 180 ◦ and 180 ◦ , so there’s a discontinuity if one tries to coordinatize the entire sphere.) We shall thus be concerned with m-dimensional manifolds M which are themselves subsets of the n-dimensional Euclidean space R n . In almost all cases we consider, m = 1, 2 ,... , or n- 1. There is a case m = 0, but these “manifolds” are zero dimensional and thus are just made up of isolated points. The case m = n is actually of some interest; however, a manifold M ⊂ R n of dimension n is just an open set in R n and is therefore essentially flat. M and R n are locally the same in this case. When M ⊂ R n we say that R n is the ambient space in which M lies. We usually call 1-dimensional manifolds “curves,” and 2-dimensional manifolds “surfaces.” But we shall generically use the neutral word “manifold.” A. Hypermanifolds Assume R n is the ambient space, and M ⊂ R n the manifold. Given that we are not very interested in the case of n-dimensional M , we distinguish manifolds which have the maximal dimension n- 1 and we call them hypermanifolds . IMPLICIT DESCRIPTION . Suppose R n g-→ R is a function which is of class C 1 . We have already thought about its level sets , sets of the form M = { x | x ∈ R n , g ( x ) = c } , where c is a constant; see p. 2–42. The fundamental thinking here is that in R n there are n independent coordinates; the restriction g ( x 1 ,...,x n ) = c removes one degree of freedom, so that points of M can locally be described in terms of only n- 1 coordinates. Thus we anticipate that M is a manifold of dimension n- 1, a hypermanifold. A very nice example of a hypermanifold is the unit sphere in R n : S (0 , 1) = { x | x ∈ R n , k x k = 1 } . 2 Chapter 5 There is a very important restriction we impose on this situation. It is motivated by our recognition from p. 2–43 that ∇ g ( x ) is a vector which should be orthogonal to M at the point x ∈ M . If M is truly ( n- 1)-dimensional, then this vector ∇ g ( x ) should probably be nonzero....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell.

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chap05 - manifolds - Manifolds 1 Chapter 5 Manifolds We are...

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