What Does it Mean for a Manifold to be Locally Euclidean?
Most recent answer: 04/09/2011
Q:
What defines what is considered "locally" Euclidean on a manifold? I know the earths curvature can be used as an example, where the earth as a whole has curvature, but locally it's flat. So my question is, what prevents local curvature and what is defined as local?
- Devon (age 24)
Lansing
- Devon (age 24)
Lansing
A:
Dear Devon,
Loosely speaking, "locally Euclidean" just means that, for every point p which lies on an n-dimensional manifold, M, the points in the neighborhood of p can be specified by using "n" coordinates (x1, x2,...,xn), and that there exists a continuous map from that neighborhood to Rn (n-dimensional Euclidean space) .
Let's take your example of the earth (a sphere), which is a 2-manifold in R3. It's "in" R3 because we live in 3-dimensional Euclidean space. It's a 2-manifold, because around every point p, you can specify p and the other points in the neighborhood by 2 dimensions, say (x1, x2), and the 3rd coordinate can be derived from the other two (e.g., z = sqrt(1-x2-y2) if x1 = x and x2 = y). Although you might be thinking, well z = EITHER sqrt(1-x2-y2) OR -sqrt(1-x2-y2). But the point is, in any specific region around a point on the sphere, say the set of points U = (points in the sphere for which z>0), we don't have that problem.
Getting even more general and even more mathematical (which I'll assume based on your follow-up question about 1-forms in E&M you can follow), an n-dimensional (sub)manifold in an (n+q)-dimensional euclidean space is defined when q coordinates of the manifold can be described differentiably in terms of the other n coordinates. Meaning, at any point p in the manifold, the neighborhood U around p (neighborhood meaning the set of points which are a certain "distance" away from p) can be described by (x1, x2,...,xn, f1(x1, x2,...,xn),...,fq(x1,x2,...,xn)).
I hope that makes sense. Definitely follow up if this is unclear!
John
Here's another take on the same idea, since different people like different ways of visualizing these things. The locally Euclidean space gets very close to Euclidean properties within very small balls. So as you look at points that get closer and closer, the sum of the angles of triangles gets closer to 180°, the ratio of the circumference of a circle to its diameter gets close to pi, etc. In all this we asume that there's a definition of distance in the space. /Mike W.
Loosely speaking, "locally Euclidean" just means that, for every point p which lies on an n-dimensional manifold, M, the points in the neighborhood of p can be specified by using "n" coordinates (x1, x2,...,xn), and that there exists a continuous map from that neighborhood to Rn (n-dimensional Euclidean space) .
Let's take your example of the earth (a sphere), which is a 2-manifold in R3. It's "in" R3 because we live in 3-dimensional Euclidean space. It's a 2-manifold, because around every point p, you can specify p and the other points in the neighborhood by 2 dimensions, say (x1, x2), and the 3rd coordinate can be derived from the other two (e.g., z = sqrt(1-x2-y2) if x1 = x and x2 = y). Although you might be thinking, well z = EITHER sqrt(1-x2-y2) OR -sqrt(1-x2-y2). But the point is, in any specific region around a point on the sphere, say the set of points U = (points in the sphere for which z>0), we don't have that problem.
Getting even more general and even more mathematical (which I'll assume based on your follow-up question about 1-forms in E&M you can follow), an n-dimensional (sub)manifold in an (n+q)-dimensional euclidean space is defined when q coordinates of the manifold can be described differentiably in terms of the other n coordinates. Meaning, at any point p in the manifold, the neighborhood U around p (neighborhood meaning the set of points which are a certain "distance" away from p) can be described by (x1, x2,...,xn, f1(x1, x2,...,xn),...,fq(x1,x2,...,xn)).
I hope that makes sense. Definitely follow up if this is unclear!
John
Here's another take on the same idea, since different people like different ways of visualizing these things. The locally Euclidean space gets very close to Euclidean properties within very small balls. So as you look at points that get closer and closer, the sum of the angles of triangles gets closer to 180°, the ratio of the circumference of a circle to its diameter gets close to pi, etc. In all this we asume that there's a definition of distance in the space. /Mike W.
(published on 04/09/2011)