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| In [[differential geometry]], '''normal coordinates''' at a point ''p'' in a [[differentiable manifold]] equipped with a [[torsion tensor|symmetric]] [[affine connection]] are a [[local coordinate system]] in a [[neighborhood (mathematics)|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 derivative]]s of the metric at ''p'' vanish.
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| 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 frame]]s. 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 manifold]]s {{harv|Busemann|1955}}.
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| ==Geodesic normal coordinates==
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| Geodesic normal coordinates are local coordinates on a manifold with an affine connection afforded by the [[exponential map]]
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| <math> \exp_p : T_{p}M \supset V \rightarrow M </math>
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| and an isomorphism
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| <math> E: \mathbb{R}^n \rightarrow T_{p}M </math>
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| given by any [[basis of a vector space|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 basis|orthonormal]], and the resulting coordinate system is then known as a '''Riemannian normal coordinate system'''.
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| 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]] ''T<sub>p</sub>M'' and exp<sub>''p''</sub> 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:
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| <math> \varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n </math>
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| 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''.
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| === Properties ===
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| The properties of normal coordinates often simplify computations. In the following, assume that ''U'' is a normal neighborhood centered at ''p'' in ''M'' and ''(x<sup>i</sup>)'' are normal coordinates on ''U''.
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| * Let ''V'' be some vector from ''T<sub>p</sub>M'' with components ''V<sup>i</sup>'' in local coordinates, and <math> \gamma_V </math> be the [[geodesic]] with starting point ''p'' and velocity vector ''V'', then <math> \gamma_V</math> is represented in normal coordinates by <math> \gamma_V(t) = (tV^1, ... , tV^n) </math> as long as it is in ''U''.
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| * The coordinates of ''p'' are (0, ..., 0)
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| * In Riemannian normal coordinates at ''p'' the components of the [[Metric tensor|Riemannian metric]] ''g'' simplify to <math> \delta_{ij} </math>.
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| * The [[Christoffel symbols]] vanish at ''p''. In the Riemannian case, so do the first partial derivatives of <math> g_{ij} </math>.
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| ==Polar coordinates==
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| On a Riemannian manifold, a normal coordinate system at ''p'' facilitates the introduction of a system of [[spherical coordinates]], known as '''polar coordinates'''. These are the coordinates on ''M'' obtained by introducing the standard spherical coordinate system on the Euclidean space ''T''<sub>''p''</sub>''M''. That is, one introduces on ''T''<sub>''p''</sub>''M'' the standard spherical coordinate system (''r'',φ) where ''r'' ≥ 0 is the radial parameter and φ = (φ<sub>1</sub>,...,φ<sub>''n''−1</sub>) is a parameterization of the [[N sphere|(''n''−1)-sphere]]. Composition of (''r'',φ) with the inverse of the exponential map at ''p'' is a polar coordinate system.
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| Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to ''p'' of nearby points. [[Gauss's lemma (Riemannian geometry)|Gauss's lemma]] asserts that the [[gradient]] of ''r'' is simply the [[partial derivative]] <math>\partial/\partial r</math>. That is,
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| :<math>\langle df, dr\rangle = \frac{\partial f}{\partial r}</math>
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| for any smooth function ''ƒ''. As a result, the metric in polar coordinates assumes a [[block diagonal]] form
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| :<math>g = \begin{bmatrix}
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| 1&0&\cdots\ 0\\
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| 0&&\\
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| \vdots &&g_{\phi\phi}(r,\phi)\\
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| 0&&
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| \end{bmatrix}.</math>
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| ==References==
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| * {{Citation | last1=Busemann | first1=Herbert | title=On normal coordinates in Finsler spaces | id={{MathSciNet | id = 0071075}} | year=1955 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=129 | pages=417–423 | doi=10.1007/BF01362381}}.
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| * {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]]|volume=Vol. 1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=0-471-15733-3}}.
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| * Chern, S. S.; Chen, W. H.; Lam, K. S.; ''Lectures on Differential Geometry'', World Scientific, 2000
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| ==See also==
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| *[[Fermi coordinates]]
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| *[[Local reference frame]]
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| [[Category:Riemannian geometry]]
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| [[Category:Coordinate systems in differential geometry]]
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