A short note to Riemann Manifold

A manifold M is like a curve in 2D and a surface in 3D. It is a set of points with topology (or neighbor structures), and locally looks like R. Example: our earth. Locally it is planar. Rigidly, for a patch U ⊂ M , we have a local coordinate system xU : U 7→ R as i = 1 · n local coordinates. Typically M can be covered by several such patches. (eg, A sphere can be covered by 2 patches but not one). In general, we omit the subscript U for clarity. Tangent space Given any point p ∈ M , it has a tangent space TpM isometric to R . If we have a metric (inner-product) in this space < ·, · >p: TpM × TpM 7→ R defined on every point p ∈ M , we thus call M Riemann Manifold. Usually (except for general relativity) the manifold M is embedded in an ambient space. For example, a 2D curve is embedded in R; our earth is embedded in R. In such cases, the tangent spaces are subspaces of the ambient space and we can have an induced inner product from ambient space. Riemann geometry aims to study the property of M without the help of ambient space (which is the typical idea of mathematicians); however, a better way to understand many important concepts is to start with a manifold with its ambient space. The union of tangent space over all the points in M is called the tangent bundle TM . A vector v ∈ TpM can be written as