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In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tangent space at ''p''. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point ''p'', thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point ''p'', and that the first partial derivatives of the metric at ''p'' vanish. A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at ''p'' only), and the geodesics through ''p'' are locally linear functions of ''t'' (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, there is no way to define normal coordinates for Finsler manifolds . ==Geodesic normal coordinates== Geodesic normal coordinates are local coordinates on a manifold with an affine connection afforded by the exponential map and an isomorphism given by any basis of the tangent space at the fixed basepoint ''p'' ∈ ''M''. If the additional structure of a Riemannian metric is imposed, then the basis defined by ''E'' may be required in addition to be orthonormal, and the resulting coordinate system is then known as a Riemannian normal coordinate system. Normal coordinates exist on a normal neighborhood of a point ''p'' in ''M''. A normal neighborhood ''U'' is a subset of ''M'' such that there is a proper neighborhood ''V'' of the origin in the tangent space ''TpM'' and exp''p'' acts as a diffeomorphism between ''U'' and ''V''. Now let ''U'' be a normal neighborhood of ''p'' in ''M'' then the chart is given by: The isomorphism ''E'' can be any isomorphism between both vectorspaces, so there are as many charts as different orthonormal bases exist in the domain of ''E''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「normal coordinates」の詳細全文を読む スポンサード リンク
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