Normal coordinates

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, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable (Busemann 1955).

Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the exponential map

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 basis 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 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 an open subset of M such that there is a proper neighborhood V of the origin in the tangent space T_pM, and exp_p acts as a diffeomorphism 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 sorts of open neighborhoods (they form a topological basis) has been established by J.H.C. Whitehead for symmetric affine connections.

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} .

• 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 geodesic 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} .
• The coordinates of the point p {\displaystyle p} are ( 0 , . . . , 0 ) {\displaystyle (0,...,0)}
• In Riemannian normal coordinates at a point p {\displaystyle p} the components of the Riemannian metric 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}} .
• The Christoffel symbols 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} .

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}).}

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_pM. That is, one introduces on T_pM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ₁,...,φ_n−1) is a parameterization of the (n−1)-sphere. 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. Gauss's lemma asserts that the gradient of r is simply the partial derivative ∂ / ∂ 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 block diagonal 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}}.}

• Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", Mathematische Annalen, 129: 417–423, doi:10.1007/BF01362381, ISSN 0025-5831, MR 0071075.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3.
• Chern, S. S.; Chen, W. H.; Lam, K. S. (2000), Lectures on Differential Geometry (hardcover ed.), World Scientific, ISBN 978-981-02-3494-2.

• Gauss Lemma
• Fermi coordinates
• Local reference frame
• Synge's world function