UDC 519.46

MATHEMATICS

G. F. KUSHNER

ON A COMPACTIFICATION OF NONCOMPACT SYMMETRIC RIEMANNIAN SPACES

(Presented by Academician I. G. Petrovskii on 10 VII 1969)

1. Let (\mathscr E) be a noncompact symmetric Riemannian space of rank (l), with maximal connected group of motions (G) and stationary subgroup (U) of some point (x_0) of it.

Denote the Lie algebras of the groups (G) and (U) by (\mathfrak G) and (\mathfrak U); denote by (\mathscr E\langle\cdot,\cdot\rangle) the invariant scalar product in the algebra (\mathfrak G) that gives rise to the Riemannian metric. The orthogonal complement (\mathscr L) of the subalgebra (\mathfrak U) in the algebra (\mathfrak G) is identified, as usual, with the tangent space of the space (\mathscr E) at the point (x_0).

Let (\mathfrak h) be some Cartan subalgebra of the space (\mathscr E) contained in (\mathscr L), (\Sigma=\Sigma(\mathfrak h)) its system of roots, and (Z) the set of integers. The set

[
I=I(\mathfrak h)=\left{\mu\in\mathfrak h\ \middle|\ \frac{2\langle\mu,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in Z\ \text{for all }\alpha\in\Sigma\right}
\tag{1}
]

will be called the lattice of the space (\mathscr E).

It is obvious that any two lattices of the space (\mathscr E) are conjugate with respect to the adjoint group (\operatorname{Ad}(G)) of the group (G).

Let ((\mathfrak h_0,\ldots,\mathfrak h_{k-1})), where (1\le k\le l), be an orthonormal root of pairwise commuting vectors of the space (\mathscr L). Choose some Cartan subalgebra (\mathfrak h) of the space (\mathscr E) so that

[
\mathfrak h_0,\ldots,\mathfrak h_{k-1}\in\mathfrak h\mathscr L,
\tag{2}
]

and select from the system of roots (\Sigma=\Sigma(\mathfrak h)) and the lattice (I=I(\mathfrak h)) the subsystem

[
\Sigma^{(k)}=\Sigma(\mathfrak h_0,\ldots,\mathfrak h_{k-1})
={\alpha\in\Sigma\mid \langle\alpha,\mathfrak h_s\rangle=0\ \text{for all }0\le s\le k-1}
\tag{3}
]

and the sublattice

[
I^{(k)}=I(\mathfrak h_0,\ldots,\mathfrak h_{k-1})
={\mu\in I\mid \langle\mu,\mathfrak h_s\rangle=0\ \text{for all }0\le s\le k-1}.
\tag{4}
]

We shall call the tuple ((\mathfrak h_0,\ldots,\mathfrak h_{k-1})) admissible if, for every (s) ((1\le s\le k-1)), (\mathfrak h_s\in((I(\mathfrak h_0,\ldots,\mathfrak h_{k-1})))) (the linear span of (I)).

1. To the space (\mathscr E) with fixed point (x_0), by means of its root system (\Sigma) and lattice (I), one can associate a family of symmetric Riemannian spaces, which we shall call the family of boundary spaces, or (\varepsilon)-spaces, of the space (\mathscr E).

An arbitrary (\varepsilon)-space of the space (\mathscr E) is constructed as follows. Let ((\mathfrak h_0,\ldots,\mathfrak h_{k-1})), where (1\le k\le l), be an arbitrary admissible tuple; let (\mathfrak h) be some Cartan subalgebra of the space (\mathscr E) satisfying (2); let (G^{(k)}=G(x_0,\mathfrak h_0,\ldots,\mathfrak h_{k-1})) be the noncompact semisimple Lie group generated by the root subsystem (3); (U^{(k)}=U(x_0,\mathfrak h_0,\ldots,\mathfrak h_{k-1})=G^{(k)}\cap U), and (H_{(k)}=H_{(k)}(x_0,\mathfrak h_0,\ldots,\mathfrak h_{k-1})) the connected noncompact commutative group corresponding to the orthogonal complement (\mathfrak h_{(k)}=\mathfrak h(x_0,\mathfrak h_0,\ldots,\mathfrak h_{k-1})) of the space (((\Sigma^{(k)}))) in the space (((I^{(k)}))) ((^{3,4})).

The direct product (\widehat{G}^{(k)}) of the groups (G^{(k)}) and (H_{(k)})

[
\widehat{G}^{(k)}=\widehat{G}(x_0,\mathfrak{h}0,\ldots,\mathfrak{h})
=G^{(k)}\cdot H_{(k)}
\tag{5}
]

does not depend on the choice of the Cartan subalgebra (\mathfrak{h}) satisfying (2), and gives rise to the symmetric Riemannian space

[
S^{(k)}=S(x_0,\mathfrak{h}0,\ldots,\mathfrak{h})
=\widehat{G}^{(k)}/U^{(k)}
\quad (1\leq k\leq l),
\tag{6}
]

which is called the (\varepsilon)-space of the (k)-th degree of the space (\mathscr{E}), corresponding to its admissible tuple ((\mathfrak{h}0,\ldots,\mathfrak{h})).

The set of all (\varepsilon)-spaces (6), supplemented for generality of the discussion by the space

[
S^{(0)}=\mathscr{E},
\tag{7}
]

is the family of (\varepsilon)-spaces of the space (\mathscr{E}).

Every (\varepsilon)-space (S^{(k)}) which does not degenerate into a point has non-positive curvature and is homeomorphic to Euclidean space. As a Riemannian space, it decomposes into the direct product

[
S^{(k)}=\mathscr{E}^{(k)}\times E^{(k)}
\tag{8}
]

of the noncompact symmetric space
[
\mathscr{E}^{(k)}=\mathscr{E}(x_0,\mathfrak{h}0,\ldots,\mathfrak{h}})=G^{(k)}/U^{(k)
]
and the Euclidean space
[
E^{(k)}=E(x_0,\mathfrak{h}0,\ldots,\mathfrak{h}/{e}})=H_{(k)
]
((e) is the identity element of the group (G)).

To each (k)-pencil (F^{(k)}) of geodesic (\varepsilon)-spaces of
[
S^{(k)}=S(x_0,\mathfrak{h}0,\ldots,\mathfrak{h})
]
((^{1})) there corresponds, in a one-to-one manner, the (\varepsilon)-space

[
S^{(k)}\bigl(x_0^{(k)},F^{(k)}\bigr)
=
S(x_0,\mathfrak{h}0,\ldots,\mathfrak{h}_k),},\mathfrak{h
\tag{9}
]

where (\mathfrak{h}_k) is the tangent vector with which the geodesic (\gamma^{(k)}\in F^{(k)}) passes through the point (x_0^{(k)}) with stationary subgroup (U^{(k)}). The space (9) is called the (\varepsilon)-space of the (k)-pencil (F^{(k)}) of geodesics of the space (S^{(k)}).

1. From the set of all projective representations of the motion group (G) of the space (\mathscr{E}), one can single out a subset (\theta) of its so-called admissible representations, or (\partial)-representations. A real projective representation (\varphi) of the group (G) is called its (\partial)-representation if in its space (P_\varphi) there exists a point (\varphi(x_0)) satisfying the conditions:

[
U\subseteq G(\varphi,x_0),\qquad
P_\varphi=P(\varphi,x_0),
\tag{10}
]

where (G(\varphi,x_0)) is the stationary subgroup of the point (\varphi(x_0)) in the group (G), and (P(\varphi,x_0)) is the projective linear manifold of the space (P_\varphi) generated by the orbit (\mathscr{E}\varphi) of the group (G) passing through the point (\varphi(x_0)). We note that the space (\mathscr{E}\varphi) is homeomorphic to the space (\mathscr{E}). If the (\partial)-representation (\varphi) satisfies the equality (U=G(\varphi,x_0)), then the space (\mathscr{E}\varphi) is isomorphic to the space (\mathscr{E}) and is called a projective representation of the space (\mathscr{E}), while its closure (\overline{\mathscr{E}}\varphi) in the space (P_\varphi) is called a projective compactification ((^{2,3})).

1. Let
   [
   P_\theta-\prod_{\varphi\in\theta} P_\varphi
   ]
   be the direct product of the spaces of all (\partial)-representations of the motion group (G) of the space (\mathscr{E}), endowed with the Tikhonov topology. It is obvious that (P_\theta) is a compact metrizable space.

Let (p=(p_\varphi){\varphi\in\theta}) be an arbitrary point of the space (P\theta), and let
[
\theta(x_0)=(\varphi(x_0))_{\varphi\in\theta}
]
(see 3). It is not hard to show that the formula

[
\theta(g)p=(\varphi(g)p_\varphi){\varphi\in\theta}
\quad (p\in P\theta,\ g\in G)
\tag{11}
]

turns the group (G) into a topological group of transformations of the space (P_\theta), in which the stationary subgroup (G(\theta,x_0)) of the point (\theta(x_0)) coincides

with stationary subgroup (U) of the point (x_0). It follows from this that the orbit (\mathscr E_\theta) of the group (G), passing through the point (\theta(x_0)), is a representation of the space (\mathscr E), and its closure (\overline{\mathscr E}\theta) in the space (P\theta) is a compactification of this space.

For any (e)-space (6) and its factors (8) one can also construct representations and compactifications in the space (P_\theta). This is done as follows. In the space (P_\theta), by means of a chain of limiting transitions,

[
\theta(x'0)=\lim)}\theta(\exp \mathfrak h_0t)\theta(x_0),\ldots,\theta(x_0^{(k)})=\lim_{t\to\infty}\theta(\exp \mathfrak h_{k-1}t)\theta(x_0^{(k-1)
]

one constructs the point (\theta(x_0^{(k)})). The orbits (S_\theta^{(k)}, \mathscr E_\theta^{(k)}, E_\theta^{(k)}) passing through it of the groups (\overline G^{(k)}, G^{(k)}, H_{(k)}) are isomorphic to the spaces (S^{(k)}, \mathscr E^{(k)}, E^{(k)}) and are called their representations generated by the compactification (\overline{\mathscr E}\theta); and the closures (\overline S\theta^{(k)}, \overline{\mathscr E}\theta^{(k)}, \overline E\theta^{(k)}) of these orbits in the space (P_\theta) are compactifications of these spaces, also generated by the compactification (\overline{\mathscr E}_\theta).

Let us describe the structure of the boundary (\mathscr T_\theta=\overline{\mathscr E}\theta\setminus\mathscr E\theta) of the compactification (\overline{\mathscr E}_\theta) and some of its connections with the geometry of the space (\mathscr E). Let (S^{(k)}) ((k\ge 0)) be an arbitrary (e)-space of the space (\mathscr E) not degenerating to a point ((^{6,7})); let (\gamma^{(k)}) be a geodesic of the space (S^{(k)}); let (\mathfrak h_k) be the tangent vector of the geodesic (\gamma^{(k)}) at some point (x_0^{(k)}), and let (\exp \mathfrak h_k t) ((-\infty<t<+\infty)) be the group of translations of the space (S^{(k)}) along the geodesic (\gamma^{(k)}). The point

[
\theta(x_{\gamma(k)})=\lim_{t\to+\infty}\theta(\exp \mathfrak h_k t)\theta(x_0^{(k)})
\tag{12}
]

is naturally called the improper point of the geodesic (\gamma^{(k)}) in the compactification (\overline S_\theta^{(k)}).

Theorem 1. All geodesics belonging to one and the same (\mathfrak h)-bundle of the (e)-space (S^{(k)}) (1) have, in the compactification (\overline S_\theta^{(k)}), a common improper point.

Let (F^{(k)}) be an arbitrary (k)-bundle of geodesics of the space (S^{(k)}) (1), and let (S^{(k)}(x_0^{(k)},F^{(k)})) be the corresponding (e)-space (9). We shall call the representation (S_\theta^{(k)}(x_0^{(k)},F^{(k)})) of the space (S^{(k)}(x_0^{(k)},F^{(k)})), generated by the compactification (\overline{\mathscr E}_\theta), the component of the boundary corresponding to the (k)-bundle (F^{(k)}).

Theorem 2. The boundary component
[
S_\theta^{(k)}(F^{(k)})=S_\theta^{(k)}(x_0^{(k)},F^{(k)})
]
of any (k)-bundle (F^{(k)}) of geodesics of the space (S^{(k)}) does not depend on the choice of the initial point (x_0^{(k)}) and coincides with the set of improper points of all geodesics of this (k)-bundle in the compactification (\overline S_\theta^{(k)}). It decomposes into the direct product

[
S_\theta^{(k)}(F^{(k)})=\mathscr E_\theta^{(k)}(x_0^{(k)},F^{(k)})\times E_\theta^{(k)}(x_0^{(k)},F^{(k)})
\tag{13}
]

of representations of the noncompact and Euclidean factors of the (e)-space (S^{(k)}(x_0^{(k)},F^{(k)})), and its foliations, generated by this decomposition, do not depend on the choice of the initial point (x_0^{(k)}).

Let (\Phi^{(k)}) be the set of all (k)-bundles of the (e)-space (S^{(k)}) ((k\ge 0)), and let (\mathscr T_\theta^{(k)}=\overline S_\theta^{(k)}/S_\theta^{(k)}).

Theorem 3. For any (e)-space (S^{(k)}) ((k\ge 0)) not degenerating to a point,

[
\mathscr T_0^{(k)}=\bigcup_{F^{(k)}\in\Phi^{(k)}}\overline S_\theta^{(k)}(F^{(k)}),
\tag{14}
]

where

[
\overline{S}{\theta}^{(k)}\bigl(F\bigr)\cap}^{(k)
\overline{S}{\theta}^{(k)}\bigl(F\bigr)=\varnothing}^{(k)
\quad \text{for } F_{1}^{(k)}\ne F_{2}^{(k)} .
\tag{15}
]

It follows directly from Theorem 3 that

Theorem 4. The family of representations of all (k)-spaces of the space (\mathscr E) of nonzero degrees (6), generated by the compactification (\mathscr E_{\theta}), is a decomposition of its boundary (\mathcal T_{\theta}).

5. Let us compare the compactification (\mathscr E_{\theta}) with the compactification (\overline{\mathscr E}) of F. I. Karpelevich ({}^{1}).

Theorem 5. For any noncompact symmetric Riemannian space (\mathscr E), the natural mapping (\theta^{-1}:\mathscr E_{\theta}\to\mathscr E) can be extended to a continuous surjection (\overline{\theta}^{-1}:\overline{\mathscr E}_{\theta}\to\overline{\mathscr E}).

Theorem 6. If the rank of the space (\mathscr E) is greater than 1, then it is impossible to extend the natural mapping (\theta:\mathscr E\to\mathscr E_{\theta}) to a continuous mapping (\overline{\theta}:\overline{\mathscr E}\to\overline{\mathscr E}_{\theta}).

Remark 1. For spaces (\mathscr E) of rank 0 or 1, the compactifications (\mathscr E_{\theta}) and (\overline{\mathscr E}) are equivalent.

Remark 2. From Theorems 5 and 6 it is clear that the compactification (\overline{\mathscr E}_{\theta}) has a more developed boundary than the compactification (\overline{\mathscr E}).

The author thanks P. K. Rashevskii for posing the problem and for guidance in its solution.

Astrakhan
State Pedagogical Institute

Received
2 VI 1969

REFERENCES

({}^{1}) F. I. Karpelevich, Tr. Mosk. matem. obshch., 14, 48 (1965).
({}^{2}) P. K. Rashevskii, Matem. sborn. 50 (92), 2, 171 (1960).
({}^{3}) I. Satake, Sborn. per. Matematika, 5, 3, 45 (1961).