# Reductive space

A homogeneous space $G/H$ of a connected Lie group $G$ such that in the Lie algebra $\mathfrak g$ of $G$ there is an $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant subspace complementary to the subalgebra $\mathfrak h \subset \mathfrak g$, where $\mathfrak h$ is the Lie algebra of the group $H$. The validity of any of the following conditions is sufficient for the homogeneous space $G/H$ to be reductive: 1) the linear group $\mathop{\rm Ad} _ {\mathfrak g} ( H)$ is completely reducible; or 2) in $\mathfrak g$ there is an $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant bilinear form whose restriction to $\mathfrak h$ is non-degenerate. In particular, any homogeneous Riemannian space is reductive. If $M = G/H$ is a reductive space and the group $G$ acts effectively on $M$, then the linear representation of the isotropy group $H$ in the tangent space $M _ {0}$ to the manifold $M$ at the point $0 = eH \in M$ is faithful (cf. Faithful representation). Two important $G$- invariant affine connections on $M$ are associated with each $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant subspace $\mathfrak m \subset \mathfrak g$ complementary to $\mathfrak h$: the canonical connection and the natural torsion-free connection. The canonical connection on the reductive space $M = G/H$ with a fixed $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant decomposition $\mathfrak g = \mathfrak h \dot{+} \mathfrak m$ is the unique $G$- invariant affine connection on $M$ such that for any vector $X \in \mathfrak m$ and any frame $u$ at the point 0, the curve $( \mathop{\rm exp} tX) u$ in the principal fibration of frames over $M$ is horizontal. The canonical connection is complete and the set of its geodesics through 0 coincides with the set of curves of the type $( \mathop{\rm exp} tX) 0$, where $X \in \mathfrak m$. After the natural identification of the spaces $\mathfrak m$ and $M _ {0}$, the curvature tensor $R$ and torsion tensor $T$ of the canonical connection are defined by the formulas $( R( X, Y) Z) _ {0} = - [[ X, Y] _ {\mathfrak h} , Z]$ and $T( X, Y) _ {0} = -[ X, Y] _ {\mathfrak m}$, where $X, Y, Z \in \mathfrak m$ and $W _ {\mathfrak h}$ and $W _ {\mathfrak m}$ denote the projections of the vector $W \in \mathfrak g$ onto $\mathfrak h$ and $\mathfrak m$, respectively.

The tensor fields $R$ and $T$ are parallel relative to the canonical connection, as is any other $G$- invariant tensor field on $M$. The Lie algebra of the linear holonomy group (see Holonomy group) of the canonical connections on $M$ with supporting point 0 is generated by the set $\{ {\lambda ([ X, Y] _ {\mathfrak h} ) } : {X, Y \in \mathfrak m } \}$, where $\lambda$ is the linear representation of the isotropy Lie algebra $\mathfrak h$ in the space $M _ {0}$. Any connected simply-connected manifold with a complete affine connection with parallel curvature and torsion fields can be represented as a reductive space whose canonical connection coincides with the given affine connection. In the reductive space $M = G/H$ with fixed $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant decomposition $\mathfrak g = \mathfrak h \dot{+} \mathfrak m$ there is a unique $G$- invariant affine connection with zero torsion having the same geodesics as the canonical connection. This connection is called the natural torsion-free connection on $M$( relative to the decomposition $\mathfrak g = \mathfrak h \dot{+} \mathfrak m$). A homogeneous Riemannian or pseudo-Riemannian space $M = G/H$ is naturally reductive if it admits an $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant decomposition $\mathfrak g = \mathfrak h \dot{+} \mathfrak m$ such that

$$\tag{* } B( X, [ Z, Y] _ {\mathfrak m} ) + B ([ Z, X] _ {\mathfrak m} , Y) = 0$$

for all $X, Y, Z \in \mathfrak m$, where $B$ is the non-degenerate symmetric bilinear form on $\mathfrak m$ induced by the Riemannian (pseudo-Riemannian) structure on $M$ under the natural identification of the spaces $\mathfrak m$ and $M _ {0}$. If $M = G/H$ is a naturally reductive Riemannian or pseudo-Riemannian space with a fixed $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant decomposition $\mathfrak g = \mathfrak h \dot{+} \mathfrak m$ that satisfies condition (*), then the natural torsion-free connection coincides with the corresponding Riemannian or pseudo-Riemannian connection on $M$. If $M$ is a simply-connected naturally reductive homogeneous Riemannian space and $M = M _ {0} \times \dots \times M _ {r}$ is its de Rham decomposition, then $M$ can be represented in the form $M = G/H$; moreover, $G = G _ {0} \times \dots \times G _ {r}$, $H = H _ {0} \times \dots \times H _ {r}$ and $M _ {i} = G _ {i} /H _ {i}$ $( i = 0 \dots r)$.

An important generalization of reductive spaces are $\nu$- reductive homogeneous spaces [4]. A homogeneous space $G/H$ is called $\nu$- reductive if its stationary subalgebra $\mathfrak h$ equals $\mathfrak h _ {1} \dot{+} \dots \dot{+} \mathfrak h _ \nu$, where $\mathfrak h _ \nu \neq \{ 0 \}$, and if there is a subspace $\mathfrak m$ in $\mathfrak h$ complementary to $\mathfrak h$ such that $[ \mathfrak h _ {i} , \mathfrak m ] \subset \mathfrak h _ {i-} 1$, $i = 1 \dots \nu$, where $\mathfrak h _ {0} = \mathfrak m$. The $1$- reductive homogeneous spaces are in fact reductive spaces; examples of $2$- reductive homogeneous spaces are projective (and conformal) spaces on which a group of projective (or conformal) transformations acts. If $M = G/H$ there is a $\nu$- reductive homogeneous space and if $\nu > 1$, then the linear representation of the isotropy Lie algebra $\mathfrak h$ is not faithful (since $[ \mathfrak h _ {i} , \mathfrak m ] \subset \mathfrak h$ when $i > 1$); consequently, there is no $G$- invariant affine connection on $M$. However, there is a canonical $G$- invariant connection on a $\nu$- reductive homogeneous space with the homogeneous space of some transitive-differential group of order $\nu$ as fibre (see [4]). Reductive and $\nu$- reductive spaces are characterized as maximally homogeneous $G$- structures (cf. $G$- structure) of appropriate type (cf. [6]).

In addition to reductive spaces, partially reductive spaces are also examined, i.e. homogeneous spaces $G/H$ such that there is a decomposition of the Lie algebra $\mathfrak g$ into a direct sum of two non-zero $\mathop{\rm Ad} _ {\mathfrak g} ( H)$- invariant subspaces, one of which contains the subalgebra $\mathfrak h$( see [5]).

#### References

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