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August 4[edit]

Definition of orientable manifolds[edit]

Let M be an n-dimensional real manifold.

Axiom A: there is a nonvanishing n-form on M.

Axiom B: M has an atlas such that for every two charts (V,f), (W,g) in this atlas and for every a in the intersection of f(V) and g(W), the determinant of the derivative of gf^-1 at a is strictly positive.

It is well known that A implies B. If M is covered by a countable atlas, then B implies A. I looked through a few standard textbooks but could not find any answer to the following

Question: Where can I find a counter example that B does not imply A?

Thank you in advance. Because it is so fundamental, I must overlook some exercises or examples in the standard textbooks. Thanks to CiaPan so that I have the following new signiture. twma 03:36, 4 August 2007 (UTC)[reply]

Please specify the definition of "manifold" that you are using. This is relevant because some definitions require that M have a countable cover; since your desired counterexample depends on this distinction, it would help to know the assumptions you're starting from. [A side note: you can render simple math expressions as e.g. gf ^-1. This speeds up the rendering of this page and widens the pool of people who can read your posts.] Tesseran 04:33, 4 August 2007 (UTC)[reply]

The following is what I understand although it may not be standard. Let M be a nonempty set. An n-patch on M is a pair (V, f) where V is a subset of M and f:V--> Rⁿ is an injection. Two n-patches (V,f),(W,g) on M are compatible if both $f(V\cap W),g(V\cap W)$ are open in Rⁿ and the coordinate transformation gf^-1 : $f(V\cap W)-->g(V\cap W)$ is a diffeomorphism, i.e. infinitely differentiable with invertible derivative. A cover AA of M by n-patches is called an n-atlas if every two members in AA are compatible. For every (V,f) in AA , since (V,f) is compatible with itself, the set f(V) is open in Rⁿ as part of the definition.

Let AA be an atlas on a set M. Let TT denote the family of subsets B of M such that for every (V,f) in AA , the set $f(B\cap V)$ is open in Rⁿ. Then TT is a topology on M. It is called the manifold topology induced by AA. A manifold is a nonempty set with an atlas so that the manifold topology is separated, i.e. distinct points have disjoint neighborhoods.

An n-patch on M is called a chart if it is compatible with every patch in AA. An atlas may contain only a few patches but we have plenty of charts to play around. In particular, a subset B of M is open iff for every m in B , there is a chart (V,f) at m with $V\subset B$ . This is frequently used as the defintion of manifold topology.

By the way, being separated is not the same as being disconnected in my mind. Paracompactness is not part of the definition of a manifold in this article. Because every manifold M is locally homeomorphic to an open subset of Rⁿ, is there any counter example to show that M as defined here is not metrizable?

Hope this would be acceptable. Thank you in advance. twma 21:12, 5 August 2007 (UTC)[reply]

Try the long line. Quoting: "The long line ... can be equipped with the structure of a (non-separable) differentiable manifold"; "[The long line] is not metrisable; this can be seen as the long ray is sequentially compact but not compact, or even Lindelöf." Tesseran 21:45, 5 August 2007 (UTC)[reply]

The long line is a manifold but not metrizable. My first original question still need help. Thank you in advance. twma 09:43, 6 August 2007 (UTC)[reply]

Overlaid maps[edit]

Suppose that there are two maps of some geographic area, to different scales. If the maps represent exactly the same area, the smaller-scale one will be correspondingly smaller than the other. If the smaller is laid on the larger, completely within it, it is intuitively obvious that there will be exactly one point where a pin can be pushed through both maps, with the marks on the maps representing the same place on the ground. But how can this point be found?…217.43.210.194 19:40, 4 August 2007 (UTC)[reply]

I'm just gonna deal with the x direction, because the y direction is the same as x. Let W and w be the widths of the larger and smaller maps, respectively, and x, 0 <= x <= W - w be the left-hand corner of the map. If x = 0 then the x-coord of the pin (call is Px) is 0, but if x = W-w (all the way over to the right) then Px is W. The function we're looking for is linear, as Px moves across the map at a proportional speed to x. So to get that, we could do it the long way with slope-intercept, or we could just intuitively say that it's Px= x * W/(W-w). Do you understand that, and does that help? Gscshoyru 19:56, 4 August 2007 (UTC)[reply]

But what if the smaller map is roatated/skewed -- SGBailey 22:31, 4 August 2007 (UTC)[reply]

No answer, but I feel someone should point out that the existence of such a point is not only intuitively obvious but also true Algebraist 22:55, 4 August 2007 (UTC)[reply]

To be as general as possible, if the smaller map's lower left corner is at (x,y) on the larger map and it has been rotated by an angle θ clockwise, but still wholly within the other, is it possible to get the co-ordinates of the point explicitly?…217.43.210.194 00:03, 5 August 2007 (UTC)[reply]

Above an appeal was made to the Brouwer fixed point theorem, which does apply here but is not constructive and does not guarantee the uniqueness of the fixed point. A less general but correspondingly more powerful theorem applies: the Banach fixed point theorem. It tells us that the fixed point is unique, and also that it is the limit of iterating the mapping that assigns, to each point on the larger map, the corresponding point on the smaller map – which is supposed to be embedded in the larger one. See also Droste effect for examples. --Lambiam 00:30, 5 August 2007 (UTC)[reply]

↑ Awesome answer. —Keenan Pepper 01:57, 5 August 2007 (UTC)[reply]

If you have a transformation that takes (x,y) on the original map to a point (a,b) on the smaller map, a scale-rotation-translation series of transformations would be something like:

a=o_{x}+s*x*cos(p)+s*y*sin(p)

b=o_{y}+s*y*cos(p)-s*x*sin(p)

Where o is the translation, s is the scale, and p is the angle. If you set a = x and b = y you turn these two equations into a solution for x and y. If you know o, s, and p, then then only unknowns left are x and y.

Alternatively, you could use a 3x3 homogeneous transformation matrix, and find its eigenvector, which should be (x,y,1)? - Rainwarrior 06:24, 5 August 2007 (UTC)[reply]