In
differential geometry, normal coordinates at a point ''p'' in a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
equipped with a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
are a
local coordinate system
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
in a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of ''p'' obtained by applying the
exponential map to the
tangent space at ''p''. In a normal coordinate system, the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
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
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
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
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
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
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
manifold. By contrast, in general there is no way to define normal coordinates for
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
s in a way that the exponential map are twice-differentiable .
Geodesic normal coordinates
Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the
exponential map
:
and an isomorphism
:
given by any
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of the tangent space at the fixed basepoint
. 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:
:
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 topological base) has been established by
J.H.C. Whitehead for symmetric affine connections.
Properties
The properties of normal coordinates often simplify computations. In the following, assume that
is a normal neighborhood centered at a point
in
and
are normal coordinates on
.
* Let
be some vector from
with components
in local coordinates, and
be the
geodesic with
and
. Then in normal coordinates,
as long as it is in
. Thus radial paths in normal coordinates are exactly the geodesics through
.
* The coordinates of the point
are
* In Riemannian normal coordinates at a point
the components of the
Riemannian metric simplify to
, i.e.,
.
* The
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
vanish at
, i.e.,
. In the Riemannian case, so do the first partial derivatives of
, i.e.,
.
Explicit formulae
In the neighbourhood of any point
equipped with a locally orthonormal coordinate system in which
and the Riemann tensor at
takes the value
we can adjust the coordinates
so that the components of the metric tensor away from
become
:
The corresponding Levi-Civita connection Christoffel symbols are
:
Similarly we can construct local coframes in which
:
and the spin-connection coefficients take the values
:
Polar coordinates
On a Riemannian manifold, a normal coordinate system at ''p'' facilitates the introduction of a system of
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, known as polar coordinates. These are the coordinates on ''M'' obtained by introducing the standard spherical coordinate system on the Euclidean space ''T''
''p''''M''. That is, one introduces on ''T''
''p''''M'' the standard spherical coordinate system (''r'',φ) where ''r'' ≥ 0 is the radial parameter and φ = (φ
1,...,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 . That is,
:
for any smooth function ''ƒ''. As a result, the metric in polar coordinates assumes a
block diagonal
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
form
:
References
* .
* {{citation , last1=Kobayashi, first1=Shoshichi, last2=Nomizu, first2=Katsumi , title =
Foundations of Differential Geometry
''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publis ...
, volume=1, publisher=
Wiley Interscience , year=1996, edition=New, isbn=0-471-15733-3.
* Chern, S. S.; Chen, W. H.; Lam, K. S.; ''Lectures on Differential Geometry'', World Scientific, 2000
See also
*
Gauss Lemma
*
Fermi coordinates
*
Local reference frame
*
Synge's world function
Riemannian geometry
Coordinate systems in differential geometry