Manifolds are one of the most fundamental concepts in Math and Physics. The informal idea of manifold is that of a space consisting of patches that "looking like locally R^n", and "smoothly sewn together". A crucial point is that the dimensionality,n, of the Euclidean space being used must be the same in every patches of the manifold; we then say that the manifold is of dimension n.
Examples of manifolds include:
- R^n itself, including the line(R), the plane (R^2), and so on.
- The n-sphere, S^n. This can be defined as the locus of all points some fixed distance from the origin in R^(n+1). The circle is of course S^1, and the two-sphere is S^2 is one of the most useful examples of manifold. We say that the S^0 is a disconnected zero-dimensional manifold. It is worth emphasizing that the definition of S^n in terms of an embedding in R^(n+1) is simply a convenient shortcut;
- The n-torus T^n results from taking an n-dimensional cube and identifying opposite sides. The two-torus T^2 is a square with opposite sides identified, as shown in Figure 2.5. The surface of a doughnut is a familiar example.
- A Riemann surface of genus g is essentially a two-torus with g holes instead of just one, as shown in Figure 2.6. For example, S^2 may be thought of as a Riemann surface of genus zero, every "compact orientable boundryless" two-dimensional manifold is a Riemann surface of some genus.
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| Note: The original picture of this picture is in the book named "An introduction to General relativity spacetime geometry by Sean M.Carrol" |
- More abstractly, a set of continuous transformation such as rotations in R^n forms a manifold. Lie groups are manifolds that also have a group structure. So for example SO(2), the set of rotation in two dimensions, is the same manifold as S^1.
- The direct product of two manifold is a manifold. This is a given manifolds M and M' of dimension n and n', we can construct a manifold MxM' of dimension n+n', consisting of ordered pairs (p,p') with p in M and p' in M'.
Final note: Here I just copied the main idea of manifold with some example from the book of Sean M.Carrol while I am reading this book for my own studying. Therefore also I would like to recommend you that once you are interested in this field may you should read that book, then you will get the more understanding about what is a manifold? and other related things.
