Symmetrical space

In mathematics , symmetric spaces are a class of Riemannian manifolds with a particularly high degree of symmetry.

They are an important class of examples in geometry and topology and are used in representation theory , harmonic analysis , number theory , modular forms, and physics, among others .

definition

A connected Riemannian manifold is a symmetric space if for each there is a reflection an , i. H. an isometry ${\displaystyle M}$ ${\displaystyle x\in M}$ ${\displaystyle x}$

${\displaystyle S_{x}:M\rightarrow M}$

With

${\displaystyle S_{x}(x)=x}$,

for their differential in ${\displaystyle x}$

${\displaystyle d_{x}S_{x}=-\mathrm {Id} }$

applies to everyone . ${\displaystyle d_{x}S_{x}(v)=-v}$ ${\displaystyle v\in T_{x}M}$

Examples

• The Euclidean space is a symmetrical space, each one defines the reflection through ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle S_{x}:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$

${\displaystyle S_{x}(y)=2x-y}$.

• The unit sphere is a symmetrical space. To is the uniquely determined point on the great circle by and , for which as well (if and are not antipodal points) applies. ${\displaystyle S^{n}\subset \mathbb {R} ^{n+1}}$ ${\displaystyle x,y\in S^{n}}$ ${\displaystyle S_{x}(y)\in S^{n}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle d(S_{x}(y),x)=d(y,x)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle S_{x}(y)\not =y}$
• Lie groups with a bi-invariant metric are symmetric spaces. The reflection is defined by ${\displaystyle S_{g}}$

${\displaystyle S_{g}(h)=gh^{-1}g}$.

Geodetic symmetry

Be a geodesic with . From follows for everyone . ${\displaystyle \gamma :\mathbb {R} \rightarrow M}$ ${\displaystyle \gamma (0)=x}$ ${\displaystyle d_{x}S_{x}=-\mathrm {Id} }$ ${\displaystyle S_{x}(\gamma (t))=\gamma (-t)}$ ${\displaystyle t\in \mathbb {R} }$

Conversely, it may be in any Riemannian manifold locally (in a sufficiently small neighborhood of a point ) is a geodetic Reflection by defined. A Riemann manifold is called locally symmetric if there is an isometry on its domain. It is a symmetrical space, if it is an isometric drawing and can be completely defined. ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle S_{x}:U\rightarrow U}$ ${\displaystyle S_{x}(\gamma (t))=\gamma (-t)}$ ${\displaystyle S_{x}}$ ${\displaystyle S_{x}}$ ${\displaystyle M}$

Homogeneous space

Every symmetrical space is a homogeneous space, i. H. of the form for a connected Lie group and a compact subgroup such that the Riemannian metric is invariant under the left effect of . Cartan characterizes symmetric spaces as follows: Let be a connected Lie group, a compact subgroup and a Lie group homomorphism with as well . (Here the fixed points of and denote the connected component of the neutral element. Is called Cartan involution .) Then carries an -invariant Riemannian metric and is a symmetrical space. ${\displaystyle G/H}$ ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle \sigma :G\rightarrow G}$ ${\displaystyle \sigma ^{2}=\mathrm {Id} }$ ${\displaystyle (G^{\sigma })_{0}\subset H\subset G^{\sigma }}$ ${\displaystyle G^{\sigma }}$ ${\displaystyle \sigma }$ ${\displaystyle (G^{\sigma })_{0}}$ ${\displaystyle \sigma }$ ${\displaystyle M=G/H}$ ${\displaystyle G}$ ${\displaystyle g}$ ${\displaystyle (M,g)}$

Cartan decomposition

Be a symmetrical space and the Cartan involution. Let be the Lie algebras of . ${\displaystyle M=G/K}$ ${\displaystyle \sigma :G\rightarrow G}$ ${\displaystyle {\mathfrak {g}},{\mathfrak {k}}}$ ${\displaystyle G,K}$

Be . Because are the only eigenvalues, the eigenspace is to the eigenvalue . We denote with the eigenspace to the eigenvalue , it corresponds to the tangent space an in . Then is and ${\displaystyle \theta =d_{0}\sigma :{\mathfrak {g}}\rightarrow {\mathfrak {g}}}$ ${\displaystyle \theta ^{2}=1}$ ${\displaystyle \pm 1}$ ${\displaystyle {\mathfrak {k}}}$ ${\displaystyle 1}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle -1}$ ${\displaystyle G/K}$ ${\displaystyle \left[e\right]}$ ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}+{\mathfrak {p}}}$

${\displaystyle [{\mathfrak {k}},{\mathfrak {k}}]\subseteq {\mathfrak {k}}}$, , . ${\displaystyle [{\mathfrak {k}},{\mathfrak {p}}]\subseteq {\mathfrak {p}}}$ ${\displaystyle [{\mathfrak {p}},{\mathfrak {p}}]\subseteq {\mathfrak {k}}}$

The shape defined using the killing shape ${\displaystyle B}$

${\displaystyle B_{\theta }(X,Y):=B(X,\theta Y)}$

is positive semidefinite.

Conversely, for a decomposition with these properties there is always an involution on that is on and on . Let be the simply connected Lie group with Lie algebra , then there is an involution with and thus a symmetric space . ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}+{\mathfrak {p}}}$ ${\displaystyle \theta }$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle +1}$ ${\displaystyle {\mathfrak {k}}}$ ${\displaystyle -1}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \theta }$ ${\displaystyle \sigma :G\rightarrow G}$ ${\displaystyle \theta =d_{0}\sigma }$ ${\displaystyle M=G/K}$

example

${\displaystyle {\mathfrak {sl}}(n,\mathbb {C} )={\mathfrak {su}}(n)\oplus {\mathfrak {p}}}$

with is a Cartan decomposition. ${\displaystyle {\mathfrak {p}}=\left\{A\in {\mathfrak {sl}}(n,\mathbb {C} ):A={\overline {A}}^{t}\right\}}$

Types of symmetrical spaces

Definitions

A symmetric space is of compact type if the killing form is on negative semidefinite. ${\displaystyle M=G/K}$ ${\displaystyle {\mathfrak {g}}}$

A symmetric space is of the Euclidean type if it is Abelian. ${\displaystyle M=G/K}$ ${\displaystyle {\mathfrak {p}}}$

A symmetric space is of the non-compact type if the killing form is non-degenerate, but not negatively semidefinite and is a Cartan decomposition. (In this case is semi-simple and a maximally compact subgroup.) ${\displaystyle M=G/K}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle G}$ ${\displaystyle K\subset G}$

Examples

• The sphere is a symmetrical space of compact type. ${\displaystyle S^{n}=SO(n+1)/SO(n)}$
• Euclidean space is a symmetrical space of Euclidean type, as is the n-dimensional torus.
• The hyperbolic space is a symmetrical space of a non-compact type. and are symmetric spaces of non-compact type. ${\displaystyle H^{n}=SO(n,1)/SO(n)}$ ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$ ${\displaystyle SL(n,\mathbb {C} )/SU(n)}$

Product disassembly

A symmetrical space is called irreducible if it cannot be decomposed as the product of two nontrivial symmetrical spaces, otherwise reducible . Every symmetrical space can be decomposed as a product of irreducible symmetrical spaces of compact, Euclidean and non-compact type.

Cutting curvature

Symmetric spaces of compact type have intersection curvature , symmetrical spaces of Euclidean type have intersection curvature , symmetrical spaces of non-compact type have intersection curvature . ${\displaystyle \geq 0}$ ${\displaystyle 0}$ ${\displaystyle \leq 0}$

Symmetrical spaces of non-compact type are CAT (0) spaces and are contractible .

duality

The symmetric space with is of compact type if and only if the symmetric space with is of non-compact type. These two symmetrical spaces are called dual to each other. At a given symmetrical space of nichtkompaktem type which is compact dual with designated. ${\displaystyle M=G_{1}/K}$ ${\displaystyle {\mathfrak {g}}_{1}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle M=G_{2}/K}$ ${\displaystyle {\mathfrak {g}}_{2}={\mathfrak {k}}\oplus i{\mathfrak {p}}\subset {\mathfrak {g}}\otimes \mathbb {C} }$ ${\displaystyle G/K}$ ${\displaystyle G^{u}/K}$

Examples:

• The hyperbolic space is dual to the sphere.
• ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$is dual to . ${\displaystyle (SO(n)\times SO(n))/SO(n)=SO(n)}$

rank

The rank of a symmetrical space is defined as ${\displaystyle M}$

${\displaystyle \mathrm {rk} (M):=\max \left\{\dim(F):F\subset T_{x}M,x\in M,\mathrm {sec} |_{F}\equiv 0\right\}}$,

d. H. the dimension of a maximum subspace on which the cutting curvature disappears.

Example: . ${\displaystyle \mathrm {rk} (SL(n,\mathbb {R} )/SO(n))=n-1}$

Symmetrical rooms of rank 1

The only non-compact symmetrical spaces with are ${\displaystyle \mathrm {rk} (M)=1}$

• ${\displaystyle \mathbb {R} ^{1}}$,
• the real hyperbolic spaces , ${\displaystyle SO(n,1)/SO(n)}$
• the complex hyperbolic spaces , ${\displaystyle SU(n,1)/SU(n)}$
• the quaternionic-hyperbolic spaces and ${\displaystyle Sp(n,1)/Sp(n)}$
• the Cayley hyperbolic plane . ${\displaystyle F_{4}^{-20}/Spin(9)}$

The only compact symmetric spaces of rank 1 are

• Spheres ,
• the real-projective spaces ,
• the complex-projective spaces,
• the quaternionic-projective spaces and
• the Cayley projective plane.

classification

There is a complete classification of symmetric spaces. In the case of compact symmetrical spaces, the following table results (for the irreducible factors into which every symmetrical space can be divided)

Label	${\displaystyle G}$	${\displaystyle K}$	dimension	rank
AI	${\displaystyle \mathrm {SU} (n)\,}$	${\displaystyle \mathrm {SO} (n)\,}$	${\displaystyle (n-1)(n+2)/2}$	${\displaystyle n-1}$
AII	${\displaystyle \mathrm {SU} (2n)\,}$	${\displaystyle \mathrm {Sp} (n)\,}$	${\displaystyle (n-1)(2n+1)}$	${\displaystyle n-1}$
AIII	${\displaystyle \mathrm {SU} (p+q)\,}$	${\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (q))\,}$	${\displaystyle 2pq}$	${\displaystyle min(p,q)}$
BDI	${\displaystyle \mathrm {SO} (p+q)\,}$	${\displaystyle \mathrm {SO} (p)\times \mathrm {SO} (q)\,}$	${\displaystyle pq}$	${\displaystyle min(p,q)}$
DIII	${\displaystyle \mathrm {SO} (2n)\,}$	${\displaystyle \mathrm {U} (n)\,}$	${\displaystyle n(n-1)}$	${\displaystyle \left[n/2\right]}$
CI	${\displaystyle \mathrm {Sp} (n)\,}$	${\displaystyle \mathrm {U} (n)\,}$	${\displaystyle n(n+1)}$	${\displaystyle n}$
CII	${\displaystyle \mathrm {Sp} (p+q)\,}$	${\displaystyle \mathrm {Sp} (p)\times \mathrm {Sp} (q)\,}$	${\displaystyle 4pq}$	${\displaystyle min(p,q)}$
EGG	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {Sp} (4)/\{\pm I\}\,}$	42	6th
EII	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {SU} (6)\cdot \mathrm {SU} (2)\,}$	40	4th
EIII	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {SO} (10)\cdot \mathrm {SO} (2)\,}$	32	2
EIV	${\displaystyle E_{6}\,}$	${\displaystyle F_{4}\,}$	26th	2
EV	${\displaystyle E_{7}\,}$	${\displaystyle \mathrm {SU} (8)/\{\pm I\}\,}$	70	7th
EVI	${\displaystyle E_{7}\,}$	${\displaystyle \mathrm {SO} (12)\cdot \mathrm {SU} (2)\,}$	64	4th
EVII	${\displaystyle E_{7}\,}$	${\displaystyle E_{6}\cdot \mathrm {SO} (2)\,}$	54	3
EVIII	${\displaystyle E_{8}\,}$	${\displaystyle \mathrm {Spin} (16)/\{\pm vol\}\,}$	128	8th
EIX	${\displaystyle E_{8}\,}$	${\displaystyle E_{7}\cdot \mathrm {SU} (2)\,}$	112	4th
FI	${\displaystyle F_{4}\,}$	${\displaystyle \mathrm {Sp} (3)\cdot \mathrm {SU} (2)\,}$	28	4th
FII	${\displaystyle F_{4}\,}$	${\displaystyle \mathrm {Spin} (9)\,}$	16	1
G	${\displaystyle G_{2}\,}$	${\displaystyle \mathrm {SO} (4)\,}$	8th	2

The classification of (irreducible) symmetric spaces of non-compact type results from the classification of compact symmetric spaces with the principle of duality .

literature

• Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34th American Mathematical Society, Providence, RI, 2001. ISBN 0-8218-2848-7
• Köhler, Kai: Differential Geometry and Homogeneous Spaces . Springer Spectrum, 2014. ISBN 978-3-8348-8313-1
• Arvanitoyeorgos, Andreas: An introduction to Lie groups and the geometry of homogeneous spaces. Translated from the 1999 Greek original and revised by the author. Student Mathematical Library, 22nd American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-2778-2
• Cheeger, Jeff; Ebin, David G .: Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. ISBN 978-0-8218-4417-5
• Helgason, Sigurdur: Geometric analysis on symmetric spaces. Second edition. Mathematical Surveys and Monographs, 39. American Mathematical Society, Providence, RI, 2008. ISBN 978-0-8218-4530-1
• Helgason, Sigurdur: Topics in harmonic analysis on homogeneous spaces. Progress in Mathematics, 13. Birkhauser, Boston, Mass., 1981. ISBN 3-7643-3051-1

Web links

• Symmetrical rooms (PDF; 466 kB)
• Homogeneous spaces (PDF; 519 kB), section 4
• Lecture Notes on Symmetric Spaces (english)
• Symmetric spaces (English)
• Groupes et géométries (French), section 4
• Differential Geometry and Symmetric Spaces (english)
• Lie groups, representation theory and symmetric spaces (English)