Why is it interesting?

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Illustration of the map from some two-dimensional manifold $M$ onto $R^2$. The neighborhood $U$ of $P$ in $M$ is mapped onto $V$ in $R^2$. This map provides a coordinate system in the neighborhood of $P$.

The standard sentence is: A Manifold is a set of points that locally "looks the same" as some Euclidean space. Such geometric spaces are particularly nice to investigate, because we can use, locally, the familiar tools from elementary calculus. Happily, such spaces are not only nice to investigate, but also useful in physics to describe nature.

The idea of locality is made precise by the mathematical concept of a neighborhood. To each point of the manifold, a neighborhood must exist that is just like a small piece of Euclidean space.

Illustration of the map from some two-dimensional manifold $M$ onto $R^2$. The neighborhood $U$ of $P$ in $M$ is mapped onto $V$ in $R^2$. This map provides a coordinate system in the neighborhood of $P$. The idea of "looks like Euclidean space" is made precise by the concept of a diffeomorphism. If there exists a map: smooth, one-to-one, onto, and with a smooth inverse, (=a diffeomorphism) for each neighborhood onto some piece of Euclidean space, the space in question is a manifold.

This map from some neighborhood $X$ of the manifold onto Euclidean space gives us local coordinates of the points in this neighborhood. The inverse map is called a parametrization of the neighborhood. Therefore another way of thinking about an n-dimensional manifold is that its a set which can be given n independent coordinates ins some neighborhood of any point.

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Examples

Example1

Example2:

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