In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the slice theorem states:
given a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
on which a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
''
''
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
as
diffeomorphisms, for any ''
'' in ''
'', the map
extends to an invariant neighborhood of
(viewed as a zero section) in
so that it defines an
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
diffeomorphism from the neighborhood to its image, which contains the orbit of ''
''.
The important application of the theorem is a proof of the fact that the quotient
admits a manifold structure when ''
'' is compact and the action is free.
In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, there is an analog of the slice theorem; it is called
Luna's slice theorem
In mathematics, Luna's slice theorem, introduced by , describes the local behavior of an action of a reductive algebraic group on an affine variety. It is an analogue in algebraic geometry of the theorem that a compact Lie group acting on a smooth ...
.
Idea of proof when ''G'' is compact
Since ''
'' is compact, there exists an invariant metric; i.e., ''
'' acts as
isometries. One then adapts the usual proof of the existence of a tubular neighborhood using this metric.
See also
*
Luna's slice theorem
In mathematics, Luna's slice theorem, introduced by , describes the local behavior of an action of a reductive algebraic group on an affine variety. It is an analogue in algebraic geometry of the theorem that a compact Lie group acting on a smooth ...
, an analogous result for
reductive algebraic group actions on
algebraic varieties
References
External links
On a proof of the existence of tubular neighborhoods*
Theorems in differential geometry
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