Given a Riemannian manifold \((M,g)\). Since the exponential map \(\exp\) is locally diffeomorph, there is a neighbourhood \(U\) of \(p\) and a neighbourhood \(V\) of \(0\in T_pM\) such that \(\exp\colon V\to U\) is diffeomorph. If \(V\) is star-shaped we call \(U\) normal neighbourhood of \(p\).
Let \((b_i)\) be an orthonormal basis on \(T_pM\), then there exists an isomorphism \(B\colon \mathbb{R}^n\to T_pM\) with \(B(x^1,\ldots,x^n)=x^ib_i\). Then \(\varphi=B^{-1}\circ (\exp_p|_V)^{-1}\colon U\to \mathbb{R}^n\) defines a smooth local chart on \(M\). It is called normal coordinates.
Remarks
- Due construction we have \(\partial_i{\mid}_p=b_i\), \(g_{ij}=\delta_{ij}\) and \(\Gamma_{ij}^k=0\) at \(p\).
- For every orthonormal basis of \(T_pM\) there is a corresponding normal coordinate system.