Normal coordinates

<h2 id="geodesic-normal-coordinates">Geodesic normal coordinates</h2>
Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the <a href="/facts/Exponential_map_(Riemannian_geometry)/YfsKa30B">exponential map</a>

exp
          
            p
          
        
        :
        
          T
          
            p
          
        
        M
        ⊃
        V
        →
        M
      
    
    {\displaystyle \exp _{p}:T_{p}M\supset V\rightarrow M}

with 
 
 
 
 V
 
 
 {\displaystyle V}
 
 an open neighborhood of 0 in 
 
 
 
 
 T
 
 p
 
 
 M
 
 
 {\displaystyle T_{p}M}
 
, and an isomorphism

E
        :
        
          
            R
          
          
            n
          
        
        →
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle E:\mathbb {R} ^{n}\rightarrow T_{p}M}

given by any <a href="/facts/Basis_of_a_vector_space/89IPoN6c">basis</a> of the tangent space at the fixed basepoint 
 
 
 
 p
 ∈
 M
 
 
 {\displaystyle p\in M}
 
. If the additional structure of a Riemannian metric is imposed, then the basis defined by E may be required in addition to be <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal</a>, 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 an open subset of M such that there is a proper neighborhood V of the origin in the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> TpM, and expp acts as a <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a> between U and V. On a normal neighborhood U of p in M, the chart is given by:

φ
        :=
        
          E
          
            −
            1
          
        
        ∘
        
          exp
          
            p
          
          
            −
            1
          
        
        :
        U
        →
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \varphi :=E^{-1}\circ \exp _{p}^{-1}:U\rightarrow \mathbb {R} ^{n}}

The isomorphism E, and therefore the chart, is in no way unique.
A convex normal neighborhood U is a normal neighborhood of every p in U. The existence of these sort of open neighborhoods (they form a <a href="/facts/Topological_basis/8EDSDVPD">topological basis</a>) has been established by <a href="/facts/J.H.C._Whitehead/NhjAGNjR">J.H.C. Whitehead</a> for symmetric affine connections.

<h3>Properties</h3>
The properties of normal coordinates often simplify computations. In the following, assume that 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a normal neighborhood centered at a point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 in 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x^{i}}
 
 are normal coordinates on 
 
 
 
 U
 
 
 {\displaystyle U}
 
.

<ul><li>Let 
 
 
 
 V
 
 
 {\displaystyle V}
 
 be some vector from 
 
 
 
 
 T
 
 p
 
 
 M
 
 
 {\displaystyle T_{p}M}
 
 with components 
 
 
 
 
 V
 
 i
 
 
 
 
 {\displaystyle V^{i}}
 
 in local coordinates, and 
 
 
 
 
 γ
 
 V
 
 
 
 
 {\displaystyle \gamma _{V}}
 
 be the <a href="/facts/Geodesic/fbWR6l80">geodesic</a> with 
 
 
 
 
 γ
 
 V
 
 
 (
 0
 )
 =
 p
 
 
 {\displaystyle \gamma _{V}(0)=p}
 
 and 
 
 
 
 
 γ
 
 V
 
 ′
 
 (
 0
 )
 =
 V
 
 
 {\displaystyle \gamma _{V}'(0)=V}
 
. Then in normal coordinates, 
 
 
 
 
 γ
 
 V
 
 
 (
 t
 )
 =
 (
 t
 
 V
 
 1
 
 
 ,
 .
 .
 .
 ,
 t
 
 V
 
 n
 
 
 )
 
 
 {\displaystyle \gamma _{V}(t)=(tV^{1},...,tV^{n})}
 
 as long as it is in 
 
 
 
 U
 
 
 {\displaystyle U}
 
. Thus radial paths in normal coordinates are exactly the geodesics through 
 
 
 
 p
 
 
 {\displaystyle p}
 
.</li>
<li>The coordinates of the point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 are 
 
 
 
 (
 0
 ,
 .
 .
 .
 ,
 0
 )
 
 
 {\displaystyle (0,...,0)}
 
</li>
<li>In Riemannian normal coordinates at a point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 the components of the <a href="/facts/Metric_tensor/tbqek1gJ">Riemannian metric</a> 
 
 
 
 
 g
 
 i
 j
 
 
 
 
 {\displaystyle g_{ij}}
 
 simplify to 
 
 
 
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle \delta _{ij}}
 
, i.e., 
 
 
 
 
 g
 
 i
 j
 
 
 (
 p
 )
 =
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle g_{ij}(p)=\delta _{ij}}
 
.</li>
<li>The <a href="/facts/Christoffel_symbols/uCyd7RWX">Christoffel symbols</a> vanish at 
 
 
 
 p
 
 
 {\displaystyle p}
 
, i.e., 
 
 
 
 
 Γ
 
 i
 j
 
 
 k
 
 
 (
 p
 )
 =
 0
 
 
 {\displaystyle \Gamma _{ij}^{k}(p)=0}
 
. In the Riemannian case, so do the first partial derivatives of 
 
 
 
 
 g
 
 i
 j
 
 
 
 
 {\displaystyle g_{ij}}
 
, i.e., 
 
 
 
 
 
 
 ∂
 
 g
 
 i
 j
 
 
 
 
 ∂
 
 x
 
 k
 
 
 
 
 
 (
 p
 )
 =
 0
 ,
 
 ∀
 i
 ,
 j
 ,
 k
 
 
 {\displaystyle {\frac {\partial g_{ij}}{\partial x^{k}}}(p)=0,\,\forall i,j,k}
 
.</li></ul>
<h3>Explicit formulae</h3>
In the neighbourhood of any point 
 
 
 
 p
 =
 (
 0
 ,
 …
 0
 )
 
 
 {\displaystyle p=(0,\ldots 0)}
 
 equipped with a locally orthonormal coordinate system in which 
 
 
 
 
 g
 
 μ
 ν
 
 
 (
 0
 )
 =
 
 δ
 
 μ
 ν
 
 
 
 
 {\displaystyle g_{\mu \nu }(0)=\delta _{\mu \nu }}
 
 and the Riemann tensor at 
 
 
 
 p
 
 
 {\displaystyle p}
 
 takes the value 
 
 
 
 
 R
 
 μ
 σ
 ν
 τ
 
 
 (
 0
 )
 
 
 {\displaystyle R_{\mu \sigma \nu \tau }(0)}
 
 we can adjust the coordinates 
 
 
 
 
 x
 
 μ
 
 
 
 
 {\displaystyle x^{\mu }}
 
 so that the components of the metric tensor away from

p
 
 
 {\displaystyle p}
 
 become

g
          
            μ
            ν
          
        
        (
        x
        )
        =
        
          δ
          
            μ
            ν
          
        
        −
        
          
            
              1
              3
            
          
        
        
          R
          
            μ
            σ
            ν
            τ
          
        
        (
        0
        )
        
          x
          
            σ
          
        
        
          x
          
            τ
          
        
        +
        O
        (
        
          |
        
        x
        
          
            |
          
          
            3
          
        
        )
        .
      
    
    {\displaystyle g_{\mu \nu }(x)=\delta _{\mu \nu }-{\tfrac {1}{3}}R_{\mu \sigma \nu \tau }(0)x^{\sigma }x^{\tau }+O(|x|^{3}).}

The corresponding Levi-Civita connection Christoffel symbols are

Γ
              
                λ
              
            
          
          
            μ
            ν
          
        
        (
        x
        )
        =
        −
        
          
            
              1
              3
            
          
        
        
          
            [
          
        
        
          R
          
            λ
            ν
            μ
            τ
          
        
        (
        0
        )
        +
        
          R
          
            λ
            μ
            ν
            τ
          
        
        (
        0
        )
        
          
            ]
          
        
        
          x
          
            τ
          
        
        +
        O
        (
        
          |
        
        x
        
          
            |
          
          
            2
          
        
        )
        .
      
    
    {\displaystyle {\Gamma ^{\lambda }}_{\mu \nu }(x)=-{\tfrac {1}{3}}{\bigl [}R_{\lambda \nu \mu \tau }(0)+R_{\lambda \mu \nu \tau }(0){\bigr ]}x^{\tau }+O(|x|^{2}).}

Similarly we can construct local coframes in which

e
          
            μ
          
          
            ∗
            a
          
        
        (
        x
        )
        =
        
          δ
          
            a
            μ
          
        
        −
        
          
            
              1
              6
            
          
        
        
          R
          
            a
            σ
            μ
            τ
          
        
        (
        0
        )
        
          x
          
            σ
          
        
        
          x
          
            τ
          
        
        +
        O
        (
        
          x
          
            2
          
        
        )
        ,
      
    
    {\displaystyle e_{\mu }^{*a}(x)=\delta _{a\mu }-{\tfrac {1}{6}}R_{a\sigma \mu \tau }(0)x^{\sigma }x^{\tau }+O(x^{2}),}

and the spin-connection coefficients take the values

ω
              
                a
              
            
          
          
            b
            μ
          
        
        (
        x
        )
        =
        −
        
          
            
              1
              2
            
          
        
        
          
            
              R
              
                a
              
            
          
          
            b
            μ
            τ
          
        
        (
        0
        )
        
          x
          
            τ
          
        
        +
        O
        (
        
          |
        
        x
        
          
            |
          
          
            2
          
        
        )
        .
      
    
    {\displaystyle {\omega ^{a}}_{b\mu }(x)=-{\tfrac {1}{2}}{R^{a}}_{b\mu \tau }(0)x^{\tau }+O(|x|^{2}).}

<h2 id="polar-coordinates">Polar coordinates</h2>
On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>, known as polar coordinates. These are the coordinates on M obtained by introducing the standard spherical coordinate system on the Euclidean space TpM. That is, one introduces on TpM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ1,...,φn−1) is a parameterization of the <a href="/facts/N_sphere/bFu2S7E2">(n−1)-sphere</a>. Composition of (r,φ) with the inverse of the exponential map at p is a polar coordinate system.
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. <a href="/facts/Gauss%27s_lemma_(Riemannian_geometry)/HaWIMsSP">Gauss's lemma</a> asserts that the <a href="/facts/Gradient/TsR7uukj">gradient</a> of r is simply the <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a> 
 
 
 
 ∂
 
 /
 
 ∂
 r
 
 
 {\displaystyle \partial /\partial r}
 
. That is,

⟨
        d
        f
        ,
        d
        r
        ⟩
        =
        
          
            
              ∂
              f
            
            
              ∂
              r
            
          
        
      
    
    {\displaystyle \langle df,dr\rangle ={\frac {\partial f}{\partial r}}}

for any smooth function ƒ. As a result, the metric in polar coordinates assumes a <a href="/facts/Block_diagonal/fPI8HX9f">block diagonal</a> form

g
        =
        
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  ⋯
                   
                  0
                
              
              
                
                  0
                
                
                
              
              
                
                  ⋮
                
                
                
                  
                    g
                    
                      ϕ
                      ϕ
                    
                  
                  (
                  r
                  ,
                  ϕ
                  )
                
              
              
                
                  0
                
                
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle g={\begin{bmatrix}1&0&\cdots \ 0\\0&&\\\vdots &&g_{\phi \phi }(r,\phi )\\0&&\end{bmatrix}}.}

<ul><li>Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>, 129: 417–423, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01362381">10.1007/BF01362381</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0071075">0071075</a>.</li>
<li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996), <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, vol. 1 (New ed.), <a href="/facts/Wiley_Interscience/53g3LqJc">Wiley Interscience</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3.</li>
<li>Chern, S. S.; Chen, W. H.; Lam, K. S. (2000), Lectures on Differential Geometry (hardcover ed.), <a href="/facts/World_Scientific/qJPTP0Yk">World Scientific</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-3494-2.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Gauss%27s_lemma_(Riemannian_geometry)/HaWIMsSP">Gauss Lemma</a></li>
<li><a href="/facts/Fermi_coordinates/C9vTVvaD">Fermi coordinates</a></li>
<li><a href="/facts/Local_reference_frame/WdBs8wgh">Local reference frame</a></li>
<li><a href="/facts/Synge%27s_world_function/KwKgbEcP">Synge's world function</a></li></ul>

Normal coordinates open-in-new

Normal coordinates