Thursday, January 21, 2010
Naive Lie Theory Note 4
S^n stands for the unit sphere in n+1 dimensional space. Perhaps this is because the circle in two dimensional space was the first "sphere" recognized making it S^1. The unit sphere is defined as the set of points a unit distance from the origin. Applying this definition to a line (one-dimensional space) would give the two points at plus one and minus one. If this had been named S^1, S^n would be the unit sphere in n-dimensional space. [I have used bold-face for the open or double letter used in the text. Also ^n stands for superscript n.]
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It's because the sphere is only the surface, not the area inside the surface as well. Therefore, you are correct, the unit circle is S1, because the circle itself is a one dimensional object. Essentially, if you are very small, walking on the suface, S1 appears to be a line, and S2 appears to be a plane.
ReplyDeleteThis is called 'Locally Euclidian in Dimension n-1'
Thanks, AnyEdge. I think I've got it.
ReplyDelete