Generalized Heisenberg groups and Damek-Ricci harmonic spaces.

*(English)*Zbl 0818.53067
Lecture Notes in Mathematics. 1598. Berlin: Springer-Verlag. viii, 125 p. (1995).

The study of generalized Heisenberg groups (groups of Heisenberg type) has its origin in the harmonic analysis of the end of the seventies. It is connected mainly with the names of A. Kaplan, F. Ricci and C. Riehm. The construction of these remarkable two-step nilpotent Lie groups with left-invariant metrics is based on the theory of Clifford modules. Very soon after these fundamental contributions to harmonic analysis had been made, it became clear that this class of Riemannian manifolds is of great interest for pure differential geometry. For instance, all geodesics can always be expressed in explicit coordinate form and also Jacobi vector fields can be expressed explicitly. The complete classification is known of all generalized Heisenberg groups for which all geodesics are orbits of one-parameter groups of isometries, or, for which all invariant differential operators commute. At this stage, the class of generalized Heisenberg groups became a source of examples and counterexamples to various conjectures in Riemannian geometry. (E.g., it provides the first examples of Riemannian manifolds whose geodesics are orbits of one- parameter groups of isometries but which are not naturally reductive – up to that time it was incorrectly believed that these two classes coincide.)

In 1987, E. Damek published her construction of semi-direct extensions of a generalized Heisenberg group by a one-dimensional real vector space which made it a solvable Lie group with a left-invariant Riemann metric. The construction is modelled according to the relationship between the solvable and nilpotent part of the Iwasawa decomposition for the isometry algebra of the complex hyperbolic space. It became clear that, for this new spaces, one can again compute geodesics in the explicit form. Also, it was shown that all these spaces are Einstein manifolds (whereas the generalized Heisenberg groups are far from being Einstein, in general). But only at the beginning of the nineties, E. Damek and F. Ricci realized that these spaces are also harmonic (i.e., the mean value of a local harmonic function over a small geodesic sphere equals the value of the function at its center). The well-known Lichnerowicz conjecture says that every harmonic space is either locally flat or locally isometric to a rank one symmetric space. The conjecture seems to be true in the compact case (this was proved by Z. Szabó under the additional condition that the fundamental group is finite). On the other hand, the Damek-Ricci spaces give a lot of noncompact harmonic spaces which are homogeneous and not locally symmetric, giving thus counterexamples to the Lichnerowicz conjecture. The problem to classify all harmonic spaces still remains open but it is now becoming much more attractive.

The purpose of the present Lecture Note is to give a comprehensive survey on generalized Heisenberg groups and Damek-Ricci spaces from the point of view of pure differential geometry. Some original results of the authors have also been included, such as the study of spectral properties of the Jacobi operators along geodesics and investigations of conjugate points in generalized Heisenberg groups. The survey starts with an overview of various classes of Riemannian manifolds which come out as natural generalizations of symmetric spaces (naturally reductive spaces, spaces with “homogeneous geodesics”, weakly symmetric spaces, commutative spaces, probabilistic commutative spaces, D’Atri spaces, Osserman spaces and others). Studying these special classes of Riemannian manifolds inside the class of generalized Heisenberg groups leads to very interesting problems, examples and charming results. (In contrast with that, the non-symmetric Damek-Ricci spaces are just harmonic and they seem to show a strong resistance to be “anything else”. This is another kind of challenge for differential geometry.)

I have full confidence that this nicely written booklet will be of great interest for everyone working in Riemannian geometry and perhaps for all differential geometers in general.

In 1987, E. Damek published her construction of semi-direct extensions of a generalized Heisenberg group by a one-dimensional real vector space which made it a solvable Lie group with a left-invariant Riemann metric. The construction is modelled according to the relationship between the solvable and nilpotent part of the Iwasawa decomposition for the isometry algebra of the complex hyperbolic space. It became clear that, for this new spaces, one can again compute geodesics in the explicit form. Also, it was shown that all these spaces are Einstein manifolds (whereas the generalized Heisenberg groups are far from being Einstein, in general). But only at the beginning of the nineties, E. Damek and F. Ricci realized that these spaces are also harmonic (i.e., the mean value of a local harmonic function over a small geodesic sphere equals the value of the function at its center). The well-known Lichnerowicz conjecture says that every harmonic space is either locally flat or locally isometric to a rank one symmetric space. The conjecture seems to be true in the compact case (this was proved by Z. Szabó under the additional condition that the fundamental group is finite). On the other hand, the Damek-Ricci spaces give a lot of noncompact harmonic spaces which are homogeneous and not locally symmetric, giving thus counterexamples to the Lichnerowicz conjecture. The problem to classify all harmonic spaces still remains open but it is now becoming much more attractive.

The purpose of the present Lecture Note is to give a comprehensive survey on generalized Heisenberg groups and Damek-Ricci spaces from the point of view of pure differential geometry. Some original results of the authors have also been included, such as the study of spectral properties of the Jacobi operators along geodesics and investigations of conjugate points in generalized Heisenberg groups. The survey starts with an overview of various classes of Riemannian manifolds which come out as natural generalizations of symmetric spaces (naturally reductive spaces, spaces with “homogeneous geodesics”, weakly symmetric spaces, commutative spaces, probabilistic commutative spaces, D’Atri spaces, Osserman spaces and others). Studying these special classes of Riemannian manifolds inside the class of generalized Heisenberg groups leads to very interesting problems, examples and charming results. (In contrast with that, the non-symmetric Damek-Ricci spaces are just harmonic and they seem to show a strong resistance to be “anything else”. This is another kind of challenge for differential geometry.)

I have full confidence that this nicely written booklet will be of great interest for everyone working in Riemannian geometry and perhaps for all differential geometers in general.

Reviewer: O.Kowalski (Praha)

##### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C30 | Differential geometry of homogeneous manifolds |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C20 | Global Riemannian geometry, including pinching |