Instead, we describe the construction of these homogeneous hypersurfaces in terms of Lie groups. We see that M has a transitive action of G if and only if it can be expressed as a coset space M =G/K. These examples are "linear" in a sense, but there are actions that do not appear to be linear.
A setM is said to be G-homogeneous if there exists a transitive action of GonM. To show the transitivity, the following is convenient. We see that G/K is G-homogeneous, and every G-homogeneous set M can be expressed as a composition space G/K. If a subgroup G′ of Gstill acts transitively on M, then M can also be expressed as a composite space M =G′/G′∩K.
Important examples of Lie groups are given by matrices, and there are two methods for determining the Lie algebras of such Lie groups. Lie algebras of Lie groups G, denoted by Lie(G), are defined as the set of all left-invariant vector fields.
The Lie algebras of particular Lie groups (1)
The Lie algebras of particular Lie groups (2)
Lie subgroups
Left-invariant metrics
Riemannian curvatures
R(X, Ei)Y, Ei⟩is called the Ricci curvature, where {Ei} is an orthonormal basis of g. 3) Kσ := ⟨R(X, Y)X, Y⟩ is called the curvature of a two-dimensional subspaceσing g, where {X, Y} is an orthonormal basis ofσ. As we saw in Example 2.11, the upper half-plane RH2 can be identified with the Lie group S. Here, as an example, we see that RH2 allows a Riemannian metric with a negative constant cross-sectional curvature, using the Lie algebra Lie(S).
Xn−1} is the Lie algebra whose bracket products are given by [A, Xi] =Xi and [Xi, Xj] = 0 (this is called the Lie algebra of the real hyperbolic space).
Second fundamental forms and shape operators
Recall that the upper half-plane RH2 = SL2(R)/SO(2) can be identified with a (solvable) Lie group. In this section, we see that the complex hyperbolic space CHn can also be identified with a certain solvable Lie group. We see that CHn can be expressed as a solvable Lie group S, which is a Lie subgroup of SU(1, n).
This concludes that every gk coincides with the k-eigenspace and (5.1) is the eigenspace decomposition of su(1,2) with respect to ad2A0.
Structure of the solvable model
Geometric structures on CHn, the Riemannian metric and the complex structure, can be translated into geometric structures on S. In fact, these geometric structures are left-invariant, hence we get structures on the Lie algebras. One can check that the almost complex structure is integrable, and that the holomorphic cross-section curvatures are −1, which is constant.
6 Homogeneous hypersurfaces in the complex hyperbolic spaces Homogeneous hypersurfaces in CHn were classified in [6]. In this section, we do not give a proof of the classification, but explain how to construct these homogeneous hypersurfaces. Recall that CHn can be expressed as solvable Lie groupsS, and the Lie algebras= Lie(S) have a basis{A0, Xj, Yj, Z0|j = 1,.
We call S′.o the corresponding submanifold tos′, where S′ is the connected Lie subgroup of S with Lie(S′) =s′. Note thatφ is the K¨ahler angle of span{X1,cos(φ)Y1+ sin(φ)X2} with respect to the complex structureJ. Note that the submanifolds corresponding to the subalgebras (A) are the point, the total geodesic CH1 and the total geodesic CH2.
They can be seen from the Lie algebra structures in the p. N) submanifold corresponding to the (N) horosphere,. Note that all these submanifolds inCH3 can be generalized to the submanifolds inCHn (see [6] for precise definitions).
Geometry of hypersurfaces of type (N) and (S)
In particular, we see that every symmetric space of non-compact type can be identified with certain solvable Lie groups.
General theory of symmetric spaces
A pair of a Lie algebra and its subalgebra, (g,k), is called a Riemannian symmetric pair of Lie algebras if. There is a correspondence between Riemannian symmetric pairs of Lie groups and Riemannian symmetric pairs of Lie algebras, more precisely,. Note that the correspondence between (G, K) and (g,k) is not one-to-one, although we assumed Gis connected (since K is not necessarily connected).
We study their bounded root systems, and see that M can be identified with soluble Lie groups. One can think that the symmetric spaces of non-compact type are symmetric spaces that satisfy the following property:. It is known that any two maximal abelian subspaces are pairwise conjugate by the (additional) action of K.
Therefore, we have the corresponding inner product⟨,⟩ons= Lie(S) (for an explicit description of ⟨,⟩ we refer to [16]) 8.1 Homogeneous hypersurfaces without focal submanifolds. This obviously gives a homogeneous hypersurface in M. If ξ∈a or ξ∈gα for some α∈Λ, then sξ :=s⊖span{ξ} is a subalgebra, and every orbit of Sξ is a homogeneous hypersurface without focal submanifolds. The first statement is easy, since Sξ is a subgroup of codimension one of S. The second statement is very difficult to prove.
From the point of view of Lie groups and Lie algebras, the geometry of these homogeneous hypersurfaces can be explored. If M = CHn, then the hypersurface of case (1) is a horosphere, and the hypersurface of case (2) is a homogeneous minimal hypersurface with a line.
Solvable Lie groups attached to subdiagrams
Canonical extension
By our theorem, any homogeneous hypersurface in RH2 can be extended to a homogeneous hypersurface in M. This article is based on the author's intensive talks at the "International Mini Workshop on Integral Geometry and Symmetric Spaces" at Kyungpook National University, February 22–24, 2010. The author would like to thank Professor Young Jin Suh and members of Kyungpook National University for their kind hospitality.
The author is also grateful to Kazuhiro Shibuya, Shinobu Fujii, Kaname Hashimoto, Takahiro Hashinaga, Akira Kubo, Satoru Tsuchie, and Katsuya Matsusaki, who carefully read the first draft of this manuscript during the seminar. Tamaru, Cohomogeneity one works on non-compact symmetric spaces with a completely geodesic single orbit, Tˆohoku Math.
