Curvature homogeneous spaces with a solvable Lie ... - Project Euclid

J. Math. Soc. Japan Vol. 44, No. 3, 1992

Curvature homogeneous spaces with a solvable Lie group as homogeneous model By Old\v{r}ich KOWALSKI, Franco TRICERRI and Lieven VANHECKE (Received July 31, 1991)

Introduction. In this paper $(M, g)$ will denote a curvature tensor $R$ defined by

$C^{\infty}$

Riemannian manifold with Riemann

$R_{XY}=D_{[X.Y]}-[D_{X}, D_{Y}]$

denotes the Levi Civita connection and $X,$ $Y$ are tangent vector fields. $(M, g)$ is said to be curvature homogeneous if for each pair of points and between the tangent spaces $T_{p}M$ and $T_{q}M$ there exists a (linear) isometry such that

where

$D$

$p$

$F$

$q$

$F^{*}(R_{q})=R_{p}$

,

or equivalently, there exists an orthonormal basis of $T_{p}M$ and one of and are the same. This that the corresponding components of $R_{p}$

$R_{q}$

$T_{q}M$

such

turns out

of $(M, g)$ to be equivalent to the following: the Riemann curvature tensor viewed as an equivariant map from the total space of the orthonormal frame bundle $OM$ of $(M, g)$ into the space $R(V)$ of algebraic curvature tensors over $V=R^{n},$ $n=\dim M$, maps $OM$ into a single $O(n)$ -orbit of $R(V)$ . (See [7] for $(M’, g’)$ more details.) Further, a homogeneous Riemannian space with Riemann curvature tensor $R’$ is called a homogeneous model for $(M, g)$ if $R(OM)\subset$ $R$

$R’(OM’)$

.

The notion of a curvature homogeneous Riemannian space has been introduced by Singer in 1960 [13]. Sekigawa [12] and Takagi [17] Produced the first examples of irreducible complete Riemannian manifolds which are curvature homogeneous but not locally homogeneous. All these examples have a symmetric model, $i.e.$ , their Riemann curvature tensor satisfies $R_{XY}\cdot R=0$

act as derivations on . Riemannian manifolds satiswhere the operators fying the last algebraic condition are said to be semi-symmetric. Many examples $R_{XY}$

$R$

This work was partially supported by contract Nr. 91.01339.01, C. N. R., Italy.

462

O. KOWALSKI, F. TRICERRI and L. VANHECKE

which are not curvature homogeneous are known. This broader class of all semi-symmetric spaces has been studied extensively by Szab\’o [14], [15], [16]. A pure existence theorem in [15] shows that the semi-symmetric spaces in each dimension depend on arbitrary functions of two variables and functions of one variable. Thus one can expect more examples of curvature homogeneous spaces inside this class. Motivated by Szab\’o’s results, and by a conjecture of M. Gromov (see [2], [6], [7], [18], [19] for more details), the authors constructed in [6], [7] new explicit examples of manifolds of the previous type which are curvature homogeneous. Also a complete study of the isometry classes is given there. These examples are obtained in a geometric way by deforming a flat right invariant metric on a semidirect product of $R$ and . In this paper we construct new examples of curvature homogeneous spaces by deforming a flat right invariant metric on a Lie group which is a semidirect product of . In this way we obtain now spaces which are curvature and homogeneous but not semi-symmetric. All of our examples are non-compact. Actually, the only known examples of compact curvature homogeneous Riemannian spaces which are not locally homogeneous are the non-homogeneous examples of isoparametric hypersurfaces in spheres described in [4]. It would be interesting to find other examples in case they exist. In this context it is worthwhile to mention some recent results of K. Yamato on three-dimensional curvature homogeneous spaces [21]. Some of his results suggest that it could be very difficult to Produce comPact three-dimensional examples, but he was able to construct new complete irreducible curvature homogeneous metrics on with three distinct principal Ricci curvatures. We shall show here that all these examples have homogeneous models. It is also remarkable that the isoparametric hypersurfaces in real space forms are not the only curvature homogeneous hypersurfaces. K. Tsukada constructed in [20] an example of a four-dimensional hypersurface in the hyperbolic space $H^{5}(-1)$ which is neither locally homogeneous nor isoParametric. , endowed with a This manifold is isometric to $(SO(3)/K)\cross R^{+},$ suitable cohomogeneity one Riemannian metric. The embedding of this manifold in $H^{6}(-1)$ has type number two and it is uniquely determined up to local congruence. We shall show that this example does not admit a homogeneous model. To our knowledge, this is the first example of a curvature homogeneous space without any homogeneous model. The paper is organized as follows. In Section 1 and Section 2 we construct our new examples. We focus on some of their curvature properties in Section 3. Section 4 is devoted to the study of the homogeneous model spaces. After giving a summary of the main results in Section 5, we concentrate on the irre$\geqq 3$

$R^{n-1}$

$R^{p}$

$R^{q}$

$R^{3}$

$K=Z_{2}\cross Z_{2}$

Curvature homogeneous spaces with a solvable Lie group

463

ducibility and the completeness of a special class of examples of Section 6. In Section 7 we treat the isometry classes of these examples. We finish this paper by giving some new results about Yamato’s and Tsukada’s examples in Section 8.

1. The group

$R^{p}\ltimes R^{q}$

with the flat invariant metric

A (connected) Lie group admits a flat left or right if and only if its Lie algebra is the semidirect metric orthogonal Abelian subalgebras and where acts on endomorphisms (see [9, p. 298]). In other words splits $\mathfrak{g}$

$f$

$\mathfrak{h}$

$\mathfrak{h}$

$\mathfrak{g}$

(1.1)

.

invariant Riemannian sum of two mutually

$G$

$g_{0}$

$g_{0}$

by skew-symmetric

$f$

as

$\mathfrak{g}=\mathfrak{h}\oplus f$

where is an Abelian subalgebra of and an Abelian ideal. Moreover, and are orthogonal and, for each $X$ belonging to , the oPerator $f$

$\mathfrak{h}$

$\mathfrak{h}$

$\mathfrak{g}$

$f$

$\mathfrak{h}$

(1.2)

$ad_{X}$

:

$\mathfrak{k}arrow \mathfrak{k}:Y-ad_{X}Y=[X, Y]$

is skew-symmetric with respect to . NOW, let and $q=\dim f$ . Then we can realize the universal covering group of as a semidirect Product of the two additive grouPs by cboosway. identify in the following and with and with We ing an orthonormal basis of be the $3o(q)$ -valued and of . Further, let by linear form defined on $g_{0}$

$p=\dim \mathfrak{h}$

$R^{p}$

$R^{p}\ltimes R^{q}$

$G$

$R^{q}$

$R^{p}$

$\mathfrak{h}$

$\mathfrak{h}$

$R^{q}$

$\mathfrak{k}$

$A$

$f$

$\mathfrak{h}=R^{p}$

(1.3)

$A(W)=-ad_{W}$ ,

Then the product in

$R^{p}\ltimes R^{q}$

$W\in b$

is given by the following rule:

$(W, X)(W’, X’)=(W+W’, e^{-A(W)}X’+X)$ .

(1.4)

and the components of the vectors $W,$ $X$ and We denote by the entries of the matrix $A(W)$ , respectively. (Here and in the sequel, the Latin , , from $p+1$ to run from 1 to , and the Greek indices indices $p+q.)$ The entries of $A$ are linear forms. Hence one may write $w^{i},$

$i,$

$x^{\alpha}$

$A_{\beta}^{\alpha}(W)$

$j$

$p$

$\alpha,$

$\beta$

$A_{\beta}^{a}$

(1.5)

$A_{\beta}^{\alpha}(W)=w^{i}A_{\iota\beta}^{\alpha}$

or

,

$A(W)=w^{i}A_{\iota}$

, . for $i=1,$ NOW, an easy computation shows that the differential one-forms

where

$A_{i}\in@o(q)$

$\cdots$

$p$

$\theta^{i}=dw^{i}$

(1.6)

,

$\{$

$\theta^{\alpha}=dx^{\alpha}+A_{t\beta}^{\alpha}x^{\beta}dw^{i}$

form a basis for the space of the right invariant differential one-forms on the . Moreover, Lie group $R^{p}\ltimes R^{q}$

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O. KOWALSKI, F. TRICERRI and L. VANHECKE

(1.7)

$g_{0}= \sum_{i=1}^{p}\theta^{i}\otimes\theta^{i}+\sum_{a=p+1}^{p+q}\theta^{\alpha}\otimes\theta^{a}$

is a flat right invariant Riemannian metric on this group. The exterior derivatives of the forms and are easily computed from (1.6). We have $\theta^{i}$

$d\theta^{i}=0$

(1.8)

$\theta^{a}$

,

$\{$

$d\theta^{\alpha}=-A_{t\beta}^{\alpha}\theta^{i}\wedge\theta^{\beta}$

.

Further, the dual basis is given by the right invariant vector fields ,

$e_{i}= \frac{\partial}{\partial w_{i}}-A_{\iota\beta}^{\alpha}x^{\beta}\frac{\partial}{\alpha}$

(1.9)

$\{$

$e_{\alpha}= \frac{\partial}{\partial x^{\alpha}}$

.

From this we get at once $[e_{\ell}, e_{j}]=0$

(1.10)

$\{$

,

$[e_{\alpha}, e_{\beta}]=0$

,

$[e_{i}, e_{\alpha}]=A_{t\alpha}^{\beta}e_{\beta}$

Finally, we note that the endomorphisms $A_{i}A_{j}=A_{j}A_{i}$

,

$i,$

.

$A_{i}$

$j=1,$

commute, $\cdots$

,

$p$

as follows immediately from the Jacobi identity and 2. The deformation of the metric

$g_{0}$

$i.e.$

,

,

(1.3).

and the curvature

homogeneous examples. NOW, (2.1)

we consider on the group manifold

$R^{p}\ltimes R^{q}$

the Riemannian metric

$g= \sum_{i=1}^{p}\omega^{i}\otimes\omega^{i}+\sum_{\alpha=p+1}^{p+q}\omega^{\alpha}\otimes\omega^{\alpha}$

where $\omega^{i}=f^{i}\theta^{i}$

(2.2)

,

$f^{i}>0$

$\{$

$\omega^{\alpha}=\theta^{\alpha}$

defined on for some positive differentiable functions normal frame is given by the vector fields $f^{i}$

$E_{t}=(f^{i})^{-1}e_{i}$ $\langle$

2.3)

$\{$

$E_{a}=e_{\alpha}$

.

,

$R^{p+q}$

. The

dual ortho-

Curvature homogeneous

spaces with a solvable Lie group

465

Then, from (2.2) and (1.8), we get $d\omega^{i}=(f^{i})^{-1}df^{i}\wedge\omega^{i}=(f^{i})^{-1}\{E_{j}(f^{i})\omega^{j}+E_{\beta}(f^{i})\omega^{\beta}\}\wedge\omega^{i}$

(2.4)

,

$\{$

$d\omega^{\alpha}=d\theta^{\alpha}=-(f^{i})^{-1}A_{\iota\beta}^{\alpha}\omega^{i}\wedge\omega^{\beta}$

.

AS usual, we compute the connection forms and the curvature forms by using the Cartan structural equations. A straightforward computation yields

$\Omega_{B}^{A}$

$\omega_{B}^{A}$

$\omega_{\beta}^{\alpha}=\sum_{t}(f^{i})^{-1}A_{\iota\beta}^{\alpha}\omega^{i}$

(2.5)

$\{$

,

$\omega_{\beta}^{i}=(f^{i})^{-1}E_{\beta}(f^{i})\omega^{i}$

,

$\omega_{j}^{i}=(f^{i})^{-1}E_{j}(f^{i})\omega^{i}-(f^{j})^{-1}E_{i}(f^{j})\omega^{j}$

,

and (2.6)

$\Omega_{\beta}^{0}=0$

(2.7)

,

$\Omega\not\in=-(f^{i})^{-1}\{\sum_{j}E_{\beta}E_{j}(f^{i})\omega^{\ell}\Lambda\omega^{j}-\sum_{\alpha}E_{\alpha}E_{\beta}(f^{\ell})\omega^{\alpha}\wedge\omega^{i}\}$

(2.8)

,

$\Omega_{j}^{i}=-(f^{i}f^{j})^{-1}g(df^{i}, df^{j})\omega^{i}\Lambda\omega^{j}$

$+(f^{i})^{-1} \sum_{k}\{E_{k}E_{j}(f^{i})-(f^{k})^{-1}E_{k}(f^{i})E_{j}(f^{k})\}\omega^{k}\wedge\omega^{i}$

$-(f^{j})^{-1} \sum_{k}\{E_{k}E_{i}(f^{j})-(f^{k})^{-1}E_{k}(f^{j})E_{i}(f^{k})\}\omega^{k}\wedge\omega^{j}$

$+(f^{i})^{-1} \sum_{a}E_{\alpha}E_{j}(f^{i})\omega^{\alpha}$

A

$\omega^{i}-(f^{f})^{-1}\sum_{\alpha}E_{\alpha}E_{i}(f^{j})\omega^{\alpha}\wedge\omega^{j}$

.

induced and Here $g(df^{i}, df^{j})$ denotes the inner product of the one-forms by the metric . Of course, the metric is in general no longer right invariant nor homogeneous. Nevertheless, it follows from (2.6), (2.7) and (2.8) that this metric is curvature homogeneous if the following sufficient conditions are satisfied: $df^{j}$

$df^{i}$

$g$

$g$

,

(2.9)

$E_{\alpha}E_{\beta}(f^{i})=\lambda_{\alpha\beta}^{i}f^{i}$

(2.10)

$E_{\beta}E_{j}(f^{i})= \mu\oint_{j}f^{i}$

(2.11)

$g(df^{i}, df^{j})=\nu^{ij}f^{i}f^{j}$

(2.12)

$(f^{i})^{-1}\{E_{k}E_{j}(f^{\ell})-(f^{k})^{-1}E_{k}(f^{i})E_{j}(f^{k})\}=\sigma_{ijk}$

for

,

,

$i\neq j$

,

for

$i\neq j$

, ,

$i\neq],$

$k\neq i$

,

and are constants. where We will not solve the system $(2.9)-(2.12)$ completely. For our purposes it will be sufficient to Provide a particular solution which Produces an interesting example. In order to do this, we note that (2.9) may be rewritten in the form $\nu^{ij}$

$\lambda_{\alpha\beta}^{i},$

(2.13)

$\mu_{\beta j}^{i},$

$\sigma_{ijk}$

$\frac{\partial^{2}f^{i}}{\partial x^{\alpha}\partial x^{\beta}}=\lambda_{\alpha\beta}^{\ell}f^{l}$

.

The integrability conditions show that the matrices one. Therefore, their entries may be written as

$\lambda^{i}=(\lambda_{\alpha\beta}^{i})$

must have rank

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KOWALSKI, F. TRICERRI and L. VANHECKE

(2.14)

$\lambda_{\alpha\beta}^{i}=\mu^{i}c_{\alpha}^{i}c_{\beta}^{i}$

where and are constant. Then (2.13) may be solved explicitly. Recall positive that we look for solutions on the whole of $R^{p+q}$ . Therefore, we may suppose that the constants are positive and equal to 1. In this case, the general solution of (2.13), and hence of (2.9), is given by $c_{\alpha}^{i}$

$\mu^{i}$

$\mu^{i}$

(2.15)

$f^{i}=a^{i}(w)e^{\Sigma_{\alpha}c_{\alpha}^{i}x^{\alpha}}+b^{i}(w)e^{-\Sigma_{\alpha}c_{a}^{i}x^{\alpha}}$

where and are arbitrary functions of $w=(w^{1}, \cdots , w^{p})$ such that In what follows we consider only the special case $a^{t}(w)$

$b^{i}(w)$

(2.16)

.

$e_{j}(f^{t})=0$

, and do not pursue the search for the general solution. for all (2.16) ditions are equivalent to $i\neq j$

$\frac{\partial a^{i}}{\partial w^{j}}-A_{f\beta}^{\alpha}x^{\beta}c_{\alpha}{}^{t}a^{i}(w)=0$

(2.17)

These con-

,

$\{$

$\frac{\partial b^{i}}{\partial w^{j}}+A_{j\beta}^{\alpha}x^{\beta}c_{\alpha}{}^{t}b^{t}(w)=0$

for all

$f^{t}>0$

$i\neq j$

,

. Hence, we must have $\frac{\partial a^{i}}{\partial w^{j}}=\frac{\partial b^{i}}{\partial w^{j}}=0$

(2.18)

,

$i\neq_{J}$

,

$\{$

$\Sigma A_{J\beta}^{\alpha}c_{\alpha}^{i}=0$

,

$i\mp]$

.

Then (2.10) and (2.12) are also satisfied and (2.11) reduces to (2.19)

$\sum_{\alpha}E_{\alpha}(f^{i})E_{\alpha}(f^{j})=\nu^{ij}f^{i}f^{j}$

,

$i\neq_{J}$

.

On the other hand, we have (2.20)

$E_{\alpha}(f^{i})=c_{\alpha}^{i}g^{i}$

,

$i=1,$

$\cdots$

,

$P$

where (2.21)

$g^{i}=a{}^{t}(w)e^{\Sigma_{\alpha}c_{\alpha}^{i}x^{a}}-b^{i}(w)e^{-\Sigma_{\alpha}c_{\alpha}^{i}x^{\alpha}}$

Therefore,

we must have

(2.22)

for

$i\neq j$

$( \sum_{\alpha}c_{\alpha}^{i}c_{\alpha}^{j})g^{i}g^{j}=\nu^{ij}f^{i}f^{f}$

. In what follows we always suppose $\sum_{\alpha}c_{a}^{f}c_{a}^{j}\neq 0$

that Then the ratios

(Note

$c_{\alpha}^{i}=0,$

$\alpha=p+1,$ $g^{i}g^{j}/f^{i}f^{j}$

.

, $p+q$ , for some gives a trivial product case.) must be constant. This is possible if and only if

$\cdots$

$i$

Curvature homogeneous

or

spaces with a solvable Lie group

identically. In the sequel we suppose $b^{i}(w)=0$ for

$a^{\ell}(u)=0$

467

$b^{i}(w)=0$

PROPOSITION 2.1. Let (2.23)

$g$

$i=1$ ,

$a^{i}(w^{i})$

If

(2.24)

is an arbitrary positive these constants satisfy $\sum_{\alpha}A_{j\beta}^{\alpha}c_{\alpha}^{i}=0$

then the metric

$g$

.. ,

$p$

. Then we get

be a metric given by (2.1), (2.2) and (1.6) such that

$f^{i}=a^{i}(w^{i})e^{\Sigma}\alpha^{c_{\alpha}^{t_{x}\alpha}}$

where each are constants.

$\cdot$

for

$i=1,$

$\cdots$

, , $p$

function of the variable

$i\neq j,$

$i,$

$j=1,$

$\cdots$

,

$p$

$w^{i}$

and the

$c_{\alpha}^{i}$

,

is curvature homogeneous.

It is useful to notice that, if we consider the vectors $c^{i}=(c_{p+1}^{i}, \cdots c’+q)$

as elements of

$R^{q}=\mathfrak{k}$

, then (2.24) is equivalent to

(2.25)

$A_{i}(c^{j})=0$

,

are the skew-symmetric operators introduced in Section 1. In where the must belong to the kernel of . for other words, the vectors Moreover, we stress the fact that we always suppose tbat the scalar pro$A_{i}$

$c^{j}$

$i\neq J$

$A_{i}$

ducts (2.26)

$\langle c^{i}, c^{f}\rangle=\sum_{\alpha}c_{\alpha}^{i}c_{\alpha}^{j}$

are different from zero. 3. Further properties of the curvature homogeneous examples. In this section we continue the study of the deformed metric under the hypotheses of Proposition 2.1. First, from (2.6), (2.7) and (2.8) we get $\Omega\beta=0$

(3.1)

,

$\{$

$\Omega\not\in=c_{\beta}{}^{t}(\sum_{\alpha}c_{\alpha}{}^{t}\omega^{\alpha})\wedge\omega^{t}$

,

$\Omega_{j}^{i}=-\langle c^{i}, c^{j}\rangle\omega^{i}\Lambda\omega^{j}$

where $\langle c^{i}, c^{j}\rangle\neq 0$

.

We use these formulas to determine the Riemann curvature tensor of is given by (3.2)

$R=-2 \sum_{A.B}\Omega_{A}^{B}\otimes\omega^{A}\wedge\omega^{B}$

,

$A,$

$B=1,$

$\cdots$

,

$p+q$

.

$g$

which

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KOwALsKI, F. TRICERRI and L. VANHECKE

We obtain (3.3)

$R=-2 \sum_{i.f}\langle c^{i}, c^{j}\rangle\omega^{i}\Lambda\omega^{f}\otimes\omega^{i}\wedge\omega^{j}-4\sum_{i.\alpha.\beta}c_{\alpha}^{i}c_{\beta}^{i}\omega^{i}\Lambda\omega^{\alpha}\mathfrak{U}^{i}\wedge\omega^{\beta}$

So, the possible

non-zero

components of

$R$

are

$R_{ifij}=-\langle c^{i}, c^{j}\rangle$

(3.4)

.

,

$i\neq j$

,

$\{$

.

$R_{i\alpha i\beta}=-c_{\alpha}^{i}c_{\beta}^{f}$

Further, it follows easily from (3.4) that the Ricci tensor (3.5)

$\rho=-\sum_{i.k}\langle C^{i}, c^{k}\rangle\omega^{i}\otimes\omega^{i}-\sum_{a.\beta,k}c_{\alpha}^{k}c\beta\omega^{\alpha}\otimes\omega^{\beta}$

and the scalar curvature

$\tau$

(3.6)

,

by $\tau=$

-II 2

$c^{i}||^{2}-F$

I

$c^{i}||^{2}$

.

we

compute the covariant derivatives with respect to the Levi connection $D$ of the $metr\dot{l}Cg$ . First, from (2.5) we have

Further,

Civita

is given by

$\rho$

$\omega_{\beta}^{o}=\sum_{t}(f^{i})^{-1}A_{l\beta}^{\alpha}\omega^{t}$

(3.7)

$\{$

$\omega\beta=c_{\beta}^{i}\omega^{i}$

$\omega_{j}^{i}=0$

,

,

.

Hence, $D_{X} \omega^{i}=-\omega^{i}(X)\sum_{\beta}c\beta\omega^{\beta}$

(3.8)

,

$\{$

$D_{X} \omega^{\alpha}=\sum_{f}c_{\alpha}^{j}\omega^{f}(X)\omega^{j}-\sum_{i.\beta}(f^{i})^{-1}A_{t\beta}^{\alpha}\omega^{i}(X)\omega^{\beta}$

and $D_{X}E_{i}=- \omega^{\ell}(X)\sum_{\beta}c_{\beta}^{i}E_{\beta}$

(3.9)

,

$\{$

$D_{X}E_{\alpha}= \sum_{j}c_{\alpha}^{j}\omega^{j}(X)E_{j}+\sum_{f.\beta}(f^{J})^{-1}A_{J}^{\beta_{\alpha}}\omega^{f}(X)E_{\beta}$

Taking into account (3.5), (3.8) and (2.24), we can of the Ricci tensor. We get covariant derivative

now

.

compute easily the

$D\rho$

(3.10)

$D \rho=\sum_{i,k.\beta}\langle c^{i}, c^{k}\rangle(c_{\beta}^{i}-c_{\beta}^{k})(\omega^{i}\otimes\omega^{\beta}\otimes\omega^{i}+\omega^{i}\otimes\omega^{i}\otimes\omega^{\beta})$

$+ \sum_{i.\alpha.\beta.\gamma}c_{\alpha}^{i}c_{\beta}^{i}A_{i\gamma}^{\alpha}(f^{i})^{-1}(\omega^{i}\otimes\omega^{\gamma}\otimes\omega^{\beta}+\omega^{i}\otimes\omega^{\beta}\otimes\omega^{\gamma})$

Therefore, the possible

non-zero

components of

$D\rho$

are

$D_{i\rho\beta i}= \sum_{k}\langle c^{i}, c^{k}\rangle(c\not\in-c_{\beta}^{k})$

(3.11)

$\{$

$D_{i}\rho_{\gamma\beta}=\Sigma c_{\alpha}^{i}c_{\beta}^{f}A_{i\mathcal{T}}^{\alpha}(f^{i})^{-1}$

$\alpha$

From this we get

.

,

.

Curvature homogeneous spaces with a solvable Lie group

469

is not PROPOSITION 3.1. Let for some $i\in\{1, , p\}$ . Then $(R^{p+q}, g)$ locally homogeneous. is not constant and the Riemannian manifold $A_{i}(c^{i})\neq 0$

$||D\rho||$

PROOF. We have (3.12)

Moreover,

$||D \rho||^{2}=2\sum_{i.\beta}(D_{i}\rho_{\beta i})^{2}+\sum_{i}||c^{i}||^{2}||A_{i}(c^{i})||^{2}(f^{i})^{-2}$

$\Sigma_{i.\beta}(D_{i}\rho_{\beta i})^{2}$

.

is constant.

does It is important to note that (3.3) shows that the curvature tensor So, only by changing on the constant . but not depend on the operators the and by keeping fixed the , we get examples of metrics on $R^{p+q}$ with the same curvature. In particular, we may put $A_{i}=0$ for all $i=1,$ , . We the Riemannian metric obtained in this way. It is defined by (2.1) denote by (2.2) and where $R$

$A_{i}$

$A_{i}$

$c_{\alpha}^{i}$

$c_{a}^{i}$

$\cdots$

$p$

$g’$

$\theta^{i}=dw^{i}$

,

$\theta^{\alpha}=dx^{\alpha}$

.

is just the direct product of In this case, the semidirect product is the standard Euclidean metric of $R^{p+q}$ . Further, from (3.8) and and are linear combinawe get that the covariant derivatives of the one-forms . From this and from (3.3) it follows tions with constant coefficients of on $R^{p+q}$ is infinitesimally homogeneous in the sense of Singer that the metric [13]. Therefore, it is locally homogeneous. Actually, it is locally isometric to a homogeneous Riemannian $space-one$ of the model $sPaces-which$ we will determine and study in the next section. $R^{p}\ltimes R^{q}$

$R^{q}$

$R^{p}$

$g_{0}$

$\omega^{A}$

$\omega^{A}\otimes\omega^{B}$

$g’$

4. The model sPaces. In contrast to what we did in Section 1, we consider now the Lie group which is a semidirect product of where is acting on as follows: $\overline{G}$

$R^{q}\ltimes R^{p}$

(4.1)

$R^{q}$

$R^{p}$

$\alpha(X)W=e^{-C(X)}W$

where

(4.2)

$C(X)=\{\begin{array}{llll}\sum_{\alpha}c_{\alpha}^{1}x^{\alpha} 0 \cdots 00 \sum_{a}c_{\alpha}^{2}x^{a} \cdots 0 0 0 \cdots \sum_{\alpha}c_{a}^{p}x^{\alpha}\end{array}\}$

The product law of (4.3)

$\overline{G}$

.

is given by

$(W_{0}, X_{0})(W, X)=(W_{0}+e^{-C(X_{0})}W, X_{0}+X)$

.

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.

KOwALsKI, F. TRICERRI and L. VANHECKE

Hence, the left translations

are given

by

$w^{i}\circ L_{(w_{0}.x_{0})}=e^{-\Sigma_{\alpha}c_{\alpha}^{l}x_{0}^{\alpha}}w^{i}+w_{0}^{i}$

(4.4)

,

$\{$

$x^{\alpha}\circ L_{(w_{0},x_{0})}=x^{\alpha}+x_{0}^{\alpha}$

.

It follows that a basis of the space of left invariant one-forms on $\overline{\omega}^{i}=\overline{f}^{i}dw^{i}$

$\langle$

4.5)

$\overline{G}$

is given by

,

$\{$

$\overline{\omega}^{\alpha}=dx^{\alpha}$

,

where (4.6)

$\overline{f}^{i}=a^{i}e^{\Sigma c^{i}x^{\alpha}}\alpha\alpha$

for some non-zero constants . The first structural equations of $a^{i}$

are

$\overline{G}$

$d\overline{\omega}^{i}=-c_{a}^{i}\overline{\omega}^{i}\Lambda\overline{\omega}^{\alpha}$

(4.7)

,

$\{$

$d\overline{\omega}^{a}=0$

Further, let

$\overline{g}$

.

be the left invariant Riemannian metric

(4.8)

$\overline{g}=\sum_{i}\overline{\omega}^{i}\otimes\overline{\omega}^{i}+\sum_{\alpha}\overline{\omega}^{\alpha}\otimes\overline{\omega}^{\alpha}$

on

$\overline{G}$

given by

.

Since (4.7) are, mutatis mutandis, exactly the same as the first structural equa, defined at the end of Section 3, tions of the locally homogeneous metric both metrics have the “same” connection forms. Namely, we have $g’$

(4.9)

$\overline{\omega}_{\beta}^{\alpha}=0$

,

with (3.7) where $A_{i}=0$ ). of the metric is given by

(compare

$\overline{\omega}_{\beta}^{i}=c_{\beta}^{i}\overline{\omega}^{i}$

,

$\overline{\omega}_{j}^{i}=0$

Hence, the Riemannian curvature tensor

$\overline{R}$

$\overline{g}$

(4.10)

$\overline{R}=-2\sum_{l.j}\langle c^{i}, c^{j}\rangle\overline{\omega}^{\ell}\wedge\overline{\omega}^{j}\otimes\overline{\omega}^{i}\wedge\overline{\omega}^{j}$

$-4 \sum_{i.\alpha.\beta}c_{\alpha}^{i}$

c’

$\overline{\omega}^{i}\wedge\overline{\omega}^{\alpha}\otimes\overline{\omega}^{i}\wedge\overline{\omega}^{\beta}$

.

Therefore, is the desired model space for all the metrics introduced in Proposition 2.1 with the corresponding constants . Note that the spaces $(R^{p+q}, g’)$ (see the end of Section 3) and are locally isometric since they have the “same” connection forms. Moreover, if the functions are bounded from below by positive constants $(\overline{G},\overline{g})$

$c_{\alpha}^{i}$

$(\overline{G},\overline{g})$

$a^{i}(w^{i})$

(4.11)

$(a^{i}(w^{i}))^{2}\geqq(a^{\ell})^{2}>0$

,

$1\leqq i\leqq p$

,

then also the metric is complete. In fact, the intrinsic distance corresponding to is bounded from below by that of (for more details, see for example[7] . In that case, $(R^{p+q}, g’)$ and are globally isometric. $g’$

$g’$

$)$

$\overline{g}$

$(\overline{G},\overline{g})$

Curvature homogeneous spaces with a solvable Lie group

Further, the covariant derivatives of the forms the Levi Civita connection of are given by

and

$\overline{\omega}^{i}$

$\overline{\omega}^{\alpha}$

471

with respect to

$\overline{D}$

$\overline{g}$

$D_{X} \overline{\omega}^{i}=-\overline{\omega}^{i}(X)\sum c_{\alpha}^{i}\overline{\omega}^{a}$

(4.12)

,

$\{$

$\overline{D}_{X}\overline{\omega}^{\alpha}=\sum c_{\alpha}^{j}\overline{\omega}^{j}(X)\overline{\omega}^{j}$

.

$j$

Using (4.12) and (4.10), a straightforward computation yields the following expression for the covariant derivative of the curvature tensor: (4.13)

$\overline{D}R=4\sum_{i.j.\beta}\langle c^{i}, c^{j}\rangle(c\not\in-c_{\beta}^{j})\overline{\omega}^{i}\otimes\{\overline{\omega}^{\beta}\wedge\overline{\omega}^{f}\mathfrak{U}^{i}\wedge\overline{\omega}^{j}+\overline{\omega}^{i}\wedge\overline{\omega}^{j}\otimes\overline{\omega}^{\beta}\Lambda\overline{\omega}^{j}\}$

Because we obtain

$\langle c^{i}, c^{j}\rangle\neq 0$

for all

$i,$

.

according to our assumption (see Section 2),

$j$

PROPOSITION 4.1. If the vectors is not symmetric.

$c^{i}$

are not all equal, then the model space

$(\overline{G},\overline{g})$

REMARK. (4.3) also implies the following general statement: A model space is symmetric if and only if the following condition holds: for each pair $(i, j)$ with $i