View source for Slice theorem

A theorem reducing the description of the action of a transformation group on some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201501.png" /> of a given [[Orbit|orbit]] to that of the [[Stabilizer|stabilizer]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201502.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201503.png" /> of this orbit on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201504.png" /> which is  "normal"  to the orbit at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201505.png" />. Namely, this theorem claims that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201506.png" /> is the homogeneous fibre space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201507.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201508.png" />. Below, the precise setting and formulation are given, together with a counterpart of the theorem for algebraic transformation groups, called the étale slice theorem.

==Slice theorem for topological transformation groups.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s1201509.png" /> be a topological [[Transformation group|transformation group]] of a [[Hausdorff space|Hausdorff space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015010.png" />. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015012.png" /> is called a slice at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015013.png" /> if the following conditions hold:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015014.png" /> is invariant under the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015016.png" />;

ii) the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015017.png" /> of all orbits intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015018.png" /> is an open neighbourhood of the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015020.png" />;

iii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015021.png" /> is the homogeneous fibre space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015022.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015023.png" />, then the equivariant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015024.png" />, which is uniquely defined by the condition that its restriction to the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015026.png" /> is the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015027.png" /> (cf. also [[Equivariant cohomology|Equivariant cohomology]]), is a [[Homeomorphism|homeomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015028.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015029.png" />. Equivalent definitions are obtained by replacing iii) either by:

iv) there is an equivariant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015030.png" /> that is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015031.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015032.png" />; or by

v) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015033.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015035.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015036.png" />.

The slice theorem claims that if certain conditions hold, then there is a slice at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015037.png" />. The conditions depend on the case under consideration. The necessity of certain conditions is explained by the following observation.

If there is a slice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015039.png" />, then there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015041.png" /> (namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015042.png" />) such that the stabilizer of every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015043.png" /> is conjugate to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015044.png" />. In general, this property fails (e.g., take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015045.png" /> acting on the space of binary forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015046.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015048.png" /> by linear substitutions. Then the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015049.png" /> is trivial but every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015050.png" /> contains a point whose stabilizer has order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015051.png" />.)

The first case in which the validity of the slice theorem was investigated is that of a compact [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015052.png" />. In this case, it has been proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015053.png" /> is a fully regular space, then there is a slice at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015054.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015055.png" /> is a [[Differentiable manifold|differentiable manifold]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015056.png" /> acts smoothly, then at every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015057.png" /> there is a differentiable slice of a special kind. Namely, in this case there is an equivariant [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015058.png" />, being the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015059.png" />, of the normal [[Vector bundle|vector bundle]] (cf. also [[Normal space (to a surface)|Normal space (to a surface)]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015060.png" /> onto an open neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015062.png" />. The image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015063.png" /> of the fibre of this bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015064.png" /> is a slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015065.png" /> which is a smooth submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015066.png" /> diffeomorphic to a vector space.

A particular case of the slice theorem for compact Lie groups was for the first time ever proven in [[#References|[a2]]]. Then the differentiable and general versions, formulated above, were proven, respectively in [[#References|[a4]]] and [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]].

The slice theorem is an indispensable tool in the theory of transformation groups, which frequently makes it possible to reduce an investigation to simple group actions like linear ones. In particular, the slice theorem is the key ingredient in the proofs of the following two basic facts of the theory:

1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015067.png" /> is a compact [[Lie group|Lie group]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015068.png" /> a separable [[Metrizable space|metrizable space]] and there are only finitely many conjugacy classes of stabilizers of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015069.png" />, then there is an equivariant embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015070.png" /> in a Euclidean vector space endowed with an orthogonal action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015071.png" />.

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015072.png" /> be a compact Lie group acting smoothly on a connected differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015073.png" />. Then there are a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015074.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015075.png" /> and a dense open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015076.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015077.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015078.png" /> and the stabilizer of every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015079.png" /> is conjugate to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015080.png" /> which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015081.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015082.png" />.

There are versions of the slice theorem for non-compact groups. For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015083.png" /> be an algebraic complex [[Reductive group|reductive group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015084.png" /> its finite-dimensional algebraic representation, both defined over the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015085.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015086.png" /> be the [[Lie group|Lie group]] of real points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015088.png" /> a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015089.png" /> containing the connected component of identity element. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015090.png" /> be a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015091.png" />-invariant differentiable submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015092.png" />, the space of real points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015093.png" />. Then, [[#References|[a7]]], for every closed orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015094.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015095.png" /> there is an equivariant diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015096.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015097.png" />-invariant neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s12015099.png" /> in the normal vector bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150100.png" /> onto a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150101.png" />-invariant saturated neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150104.png" /> (a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150105.png" /> is saturated if the fact fact that the closure of an orbit intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150106.png" /> implies that this orbit lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150107.png" />). In this case, the image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150108.png" /> of the fibre of the natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150109.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150110.png" /> is a slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150111.png" /> for the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150112.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150113.png" />.

==Slice theorem for algebraic transformation groups.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150114.png" /> be an algebraic transformation group of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150115.png" />, all defined over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150116.png" />. It happens very rarely that the literal analogue of the slice theorem above holds in this setting. Loosely speaking, the reason is that Zariski-open sets are  "too big"  (see [[#References|[a12]]], 6.1). One obtains an algebraic counterpart of a compact Lie group action on a differentiable manifold by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150117.png" /> to be a [[Reductive group|reductive group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150118.png" /> an [[Affine variety|affine variety]]. In this setting, a counterpart of a slice is given by the notion of an étale slice, defined as follows.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150119.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150120.png" /> such that the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150121.png" /> is closed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150122.png" /> be an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150123.png" />-invariant subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150124.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150125.png" />. As above, one can consider the homogeneous fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150126.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150127.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150128.png" />, and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150129.png" />. In this situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150130.png" /> is an affine variety, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150131.png" /> is a [[Morphism|morphism]], there are the categorical quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150133.png" /> and an induced morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150134.png" />, cf. [[#References|[a12]]]. The subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150135.png" /> is called an étale slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150136.png" /> if

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150137.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150138.png" /> by means of the [[Base change|base change]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150139.png" />; and

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150140.png" /> is an [[Etale morphism|étale morphism]].

The étale slice theorem, proved in [[#References|[a6]]], claims that there is an étale slice at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150141.png" /> such that the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150142.png" /> is closed.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150143.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150144.png" />, the field of complex numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150145.png" /> is a smooth point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150146.png" />, then the étale slice theorem implies that there exists an analytic slice at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150147.png" />. More precisely, there is an invariant analytic neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150148.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150149.png" /> which is analytically isomorphic to an invariant analytic neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150150.png" /> in the normal vector bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150151.png" />, cf. [[#References|[a7]]], [[#References|[a12]]].

Like the slice theorem for topological transformation groups, the étale slice theorem is an indispensable result in the investigation of algebraic transformation groups; see [[#References|[a6]]] for some basic results deduced from this theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Bredon,   "Introduction to compact transformation groups" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Gleason,   "Spaces with a compact Lie group of transformations"  ''Proc. Amer. Math. Soc.'' , '''1'''  (1950)  pp. 35–43</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jänich,   "Differenzierbare <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120150/s120150152.png" />-Mannigfaltigkeiten" , ''Lecture Notes Math.'' , '''6''' , Springer  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Koszul,   "Sur certains groupes de transformation de Lie"  ''Colloq. Inst. C.N.R.S., Géom. Diff.'' , '''52'''  (1953)  pp. 137–142</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Koszul, J.L.,   "Lectures on groups of transformations" , Tata Inst.  (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Luna,D.,   "Slices étales"  ''Bull. Soc. Math. France'' , '''33'''  (1973)  pp. 81–105</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Luna, D.,   "Sur certaines opérations différentiables des groups de Lie"  ''Amer. J. Math.'' , '''97'''  (1975)  pp. 172–181</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Montgomery,   C.T. Yang,   "The existence of slice"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 108–116</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G.D. Mostow,   "On a theorem of Montgomery"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 432–446</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Palais,   "Embeddings of compact differentiable transformation groups in orthogonal representations"  ''J. Math. Mech.'' , '''6'''  (1957)  pp. 673–678</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.S. Palais,   "Slices and equivariant imbeddings" , ''Sem. Transformation Groups'' , Princeton Univ. Press  (1960)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  V.L. Popov,   E.B. Vinberg,   "Invariant theory" , ''Algebraic Geometry IV'' , ''Encycl. Math. Sci.'' , '''55''' , Springer  (1994)  pp. 122–284</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  Wu Yi Hsiang,   "Cohomology theory of topological transformation groups" , ''Ergebn. Math.'' , '''85''' , Springer  (1979)</TD></TR></table>