Curvature-homogeneous spaces of type (1,3)

Curvature-homogeneous spaces of type (1,3) Oldřich Kowalski (Charles University, Prague), joint work with Alena Vanžurová (Palacky University, Olomouc) Zlatibor, September 3-8, 2012

Curvature homogeneity (classical) (Pseudo-)Riemannian spaces whose curvature tensor of type (0,4) is the same at all points. Trivial examples: connected locally homogeneous manifolds. Curvature homogeneous spaces have been studied by many authors, starting from the basic paper I.M. Singer: Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697.

First non-trivial Riemannian examples were given by K. Sekigawa: On some 3-dimensional Riemannian manifolds, Hokkaido Math. J. 2 (1973), 259-270, K. Sekigawa: On some 3-dimensional curvature homogeneous spaces, Tensor 31 (1977), 87-97, and generalized later on. We will survey some older results (with E. Boeckx, F. Tricerri, L. Vanhecke) and some very fresh results (with Z. Dušek and A. Vanžurová). For example, an older result says that every irreducible Riemannian manifold with the same curvature tensor as a symmetric space and NOT locally homogeneous (so-called non-homogeneous relative of a symmetric space ) is a generalized Sekigawa example.

Generalized Sekigawa examples R n+1 [w, x 1,..., x n ], open U R 2 [w, x 1 ], f : U R (smooth) non-vanishing function, A(w) = (A i j (w)) skew-symmetric (smooth) (n n)-matrix function of one variable ω 0 = f(w, x 1 )dw, ω i = dx i + n j=1 Ai j (w)xj dw, i = 1,..., n

Metric We define a metric g f,a(w) as follows: g f,a(w) = n+1 j=0 ωj ω j on open Ũ Rn+1 curvature: R 0101 = R 1010 = R 1001 = R 0110 = f 1 f x 1 x 1, all other components R ijkl vanish. g f,a(w) is nonflat and curvature homogeneous if and only if f 1 f x 1 x 1 = k where k 0 is a constant;

Non-flat curvature homogeneous examples it means: f(w, x 1 ) = a(w) exp( kx 1 ) + b(w) exp( kx 1 ) if k > 0, or f(w, x 1 ) = a(w) cos( kx 1 ) + b(w) sin( kx 1 ) if k < 0, a(w) and b(w) differentiable functions such that f(w, x 1 ) > 0 in U. Here U can be the whole plane in the case k > 0 and an open strip in the plane for k < 0.

Recall that this class of spaces is remarkable: it includes all irreducible curvature homogeneous spaces which are not locally homogeneous and whose curvature tensor R is the same as that of a Riemannian symmetric space (so-called non-homogeneous relatives of symmetric spaces ).

References E. Boeckx, O. Kowalski, L. Vanhecke: Riemannian Manifolds of Conullity two. World Scientific, 1996. O.Kowalski, F.Tricerri, L.Vanhecke: New examples of non-homogeneous Riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space. C. R. Acad. Sci. Paris, t. 311, Série I, (1990), 355-360. L. Nicolodi and F. Tricerri: On two theorems of I.M. Singer about homogeneous spaces. Ann. Global Anal. Geom., 8 (1990), 193-209. O. Kowalski, F. Tricerri, L. Vanhecke: Curvature homogeneous Riemannian manifolds. J. Math. Pures Appl., 71 (1992) 471-501.

O. Kowalski, F. Tricerri, L. Vanhecke: Curvature homogeneous spaces with a solvable Lie group as homogeneous model. J. Math. Soc. Japan 44 (1992), 461-484. E. Boeckx, O. Kowalski, L. Vanhecke: Non-homogeneous relatives of symmetric spaces. Diff. Geom. and Appl. 4 (1994), 45-69. K. Sekigawa, H. Suga and L. Vanhecke: Four-dimensional curvature homogeneous spaces, Comment. Math. Univ. Carolinae 33 (1992), 261-268. K. Sekigawa, H. Suga and L. Vanhecke: Curvature homogeneity for four-dimensional manifolds, J. Korean Math. Soc. 32 (1995), 93-101.

Lorentzian case 3-dimensional Lorentzian case: P. Bueken: On curvature homogeneous three-dimensional Lorentzian manifolds, J. Geom. Phys. 22 (1997), 349-362. P. Bueken and M. Djorić: Three-dimensional Lorentz metrics and curvature homogeneity of order one, Ann. Glob. Anal. Geom. 18 (2000), 85-103.

Pseudo-Riemannian case Pseudo-Riemannian curvature homogeneous spaces (arbitrary dimension, signature, order): P.J. Gilkey: The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds. ICP Advanced Texts in Mathematics - Vol. 2. Imperial College Press, 2007. M. Brozos-Vásquez, P. Gilkey and S. Nikčević: Geometric realizations of curvature. Imperial College Press (in preparation).

Another topic: We introduce a new concept of a curvature-homogeneous space, namely in the sense that not the curvature tensor of type (0,4) is preserved from point to point but the curvature tensor of type (1,3) is preserved from point to point, in some sense. Each curvature-homogeneous space in the classical sense is also curvature-homogeneous in the modified sense. We give proper examples of the new spaces in all dimensions, and a complete classification of such spaces in dimension 3 and in generic case.

Fresh results: O. Kowalski, A. Vanžurová: On curvature homogeneous spaces of type (1,3). Math. Nachr. 284, 17-18 (2011), 2127-2132. O. Kowalski, A. Vanžurová: On a generalization of curvature homogeneous spaces (to appear in Results in Math.) O. Kowalski, Z. Dušek: Pseudo-Riemannian spaces modelled on symmetric spaces. Monatshefte für Math. 165, 3-4 (2012), 319-326.

Notation (M, g) a smooth Riemannian manifold with a (positive) metric g and the Riemannian (L.-C.) connection, R is the type (1,3) curvature tensor, R(X, Y )Z = [ X, Y ]Z [X,Y ] Z, R is the type (0,4) curvature tensor, R p its value in p, R ijkl are local components

Definitions Definition 1 A smooth Riemannian manifold (M, g) is said to be curvature homogeneous if, for any pair of points p and q in M, there exists a linear isometry F : T p M T q M such that F (R q ) = R p. Definition 2 A Riemannian manifold (M, g) is said to be (1,3)-curvature homogeneous if for any pair of points p, q there is a curvature-preserving linear homothety f : T p M T q M, f (R q ) = R p.

Proposition 1 If (M, g) is curvature homogeneous then it is also (1,3)-curvature homogeneous. But NOT vice versa. Examples show: curvature homogeneous spaces of type (1,3) form a much bigger class.

Proposition 2 Let (M, g) be a smooth Riemannian manifold. Then the following two conditions are equivalent: (i) For each q M, there is a linear homothety f q : T p M T q M such that R p = f q (R q ), i.e. (M, g) is (1,3)-curvature homogeneous. (ii) There is a smooth function ϕ on M such that ϕ(p) = 0 and for each q M, R p = e 2ϕ(q) F q (R q ) where F q : T p M T q M is a linear isometry. R of type (0,4), R of type (1,3).

Remarks The linear isometry F q from part (i) of Proposition 2 is not always uniquely defined, but, it is also not arbitrary, in general. It must be just compatible with the property of the composed map f as given in part (ii). Every (1,3)-curvature homogeneous space of dimension n determines a uniquely determined smooth function of n variables. The converse remains an open problem but it is true in dimension 3.

Lemma 1 Let (M, g) be a Riemannian manifold and let E 1,..., E n be an orthonormal moving frame on a domain U M. Fix a point p U. Suppose that, with respect to this moving frame, R ijkl (q) = φ(q)r ijkl (p) for each point q U and for all choices of indices, where φ(q) is a smooth and positive function on U. Then there is a smooth function ϕ(q) such that ϕ(p) = 0 and, for each point q, R p = e 2ϕ(q) F q (R q ) where F q : T p M T q M is a linear isometry.

Generalized Sekigawa s examples: R n+1 [w, x 1,..., x n ], open U R 2 [w, x 1 ], f : U R (smooth) non-vanishing function, A(w) = (A i j (w)) skew-symmetric (smooth) (n n)-matrix function of one variable, ω 0 = f(w, x 1 )dw, ω i = dx i + n j=1 Ai j (w)xj dw, i = 1,..., n n+1 g f,a(w) = ω j ω j j=0

Construction The space (R n+1, g) which is (1,3)-curvature homogeneous but not (0,4)-curvature homogeneous: f an arbitrary smooth function on R 2 s.t. at all points, f and f 1 f x 1 x are nonzero and f 1 x 1 x /f is never a constant in 1 an open domain of R 2. The corresponding metric g = g f,a(w) defined on R n+1 has the curvature components R ijkl calculated as above, satisfying R ijkl (q) = (f 1 (q)f x 1 x 1 (q))/(f 1 (p)f x 1 x 1 (p))r ijkl (p) for any pair of points p, q R n+1 and all indices i, j, k, l.

Let now the point p be fixed. The assumptions of the Lemma are satisfied, where the corresponding function φ(q) is defined as and hence positive. φ(q) = f 1 (q)f x 1 x 1 (q)/(f 1 (p)f x 1 x 1 (p)) From Proposition 2 and our special assumptions we deduce that the space (R n+1, g) is (1,3)-curvature homogeneous but not (0,4)-curvature homogeneous.

The family of all such metrics depends, up to a local isometry, on 1 arbitrary function of 2 variables and a finite number of arbitrary functions of 1 variable. Moreover, if f(w, x 1 ) is defined on the whole R n+1 and if there are two continuous functions α(x 1 ), β(x 1 ) on R n+1 such that 0 < α(x 1 ) f(w, x 1 ) β(x 1 ) <, then the space (R n+1, g f,a(w) ) is a complete Riemannian manifold for an arbitrary skew-symmetric matrix function A(w). On the other hand, if A(w) is a constant matrix, then the inequality f(w, x 1 ) > ε > 0 is sufficient for the completeness.

Dimension 3 In dimension 3, a Riemannian manifold (M, g) is curvature homogeneous if and only if the Ricci eigenvalues ϱ 1, ϱ 2, ϱ 3 are constant at all points; the curvature tensor R is uniquely determined by the corresponding Ricci tensor ϱ and metric g, R ijkl = 1 n 2 (g ikϱ jl g il ϱ jk + g jl ϱ ik g jk ϱ il ) τ + (n 1)(n 2) (g ilg jk g ik g jl ). (Metrics and functions - real analytic.) (1)

Prescribed Ricci eigenvalues The following results are known: Theorem A All real analytic Riemannian manifolds with the prescribed constant Ricci eigenvalues ϱ 1 = ϱ 2 ϱ 3 depend (up to a local isometry) on two arbitrary functions of one variable. O. Kowalski: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ϱ 1 = ϱ 2 ϱ 3. Nagoya Math. J. 132 (1993), 1-36

Theorem B All real analytic Riemannian manifolds with the prescribed distinct constant Ricci eigenvalues ϱ 1 >ϱ 2 >ϱ 3 depend (up to a local isometry) on 3 arbitrary functions of 2 variables. O. Kowalski, Z. Vlášek: Classification of Riemannian 3-manifolds with distinct constant principal Ricci curvatures. Bulletin of the Belgian Mathematical Society-Simon Stevin, 5 (1998), 59-68. The spaces (M, g) with prescribed constant Ricci eigenvalues are, with rare exceptions, not locally homogeneous.

The classification of all triplets of distinct real numbers which can be realized as Ricci eigenvalues on a 3-dimensional locally homogeneous space was made in O. Kowalski, S.Ž. Nikčević: On Ricci eigenvalues of locally homogeneous Riemannan manifolds. Geometriae Dedicata 62 (1996), 65-72.

Generalized Yamato examples The authors constructed so-called generalized Yamato examples which are explicit for each choice of the triplet ϱ 1 > ϱ 2 > ϱ 3 of prescribed Ricci eigenvalues, in O. Kowalski, F. Prüfer: On Riemannian 3-manifolds with distinct constant Ricci eigevalues. Math. Ann. 300 (1994), 17-28.

The original construction by K. Yamato - some restrictions are put on the triplets ϱ 1 > ϱ 2 > ϱ 3, all constructed metrics are complete. K. Yamato: A characterization of locally homogeneous Riemannian manifolds of dimension 3. Nagoya Math. J. 123 (1993), 77-99. Theorem B was later generalized in

Theorem C All Riemannian metrics defined in a domain U R 3 [x, y, z] with the prescribed distinct real analytic Ricci eigenvalues ϱ 1 (x, y, z) > ϱ 2 (x, y, z) > ϱ 3 (x, y, z) depend, up to a local isometry, on three arbitrary (real analytic) functions of two variables. Every solution of the problem is defined at least locally, i.e. in a neighborhood U U of a fixed p U. O. Kowalski, Z. Vlášek: On 3D-manifolds with prescribed Ricci eigenvalues. In: Complex, Contact and Symmetric Manifolds-In Honor of L. Vanhecke. Progress in Mathematics, Vol. 234, Birkhäuser Boston-Basel-Berlin, pp. 187-208 (2005).

In a domain U R 3 [x, y, z], fix a point p and choose a real analytic function ϕ(x, y, z) on U vanishing at p. According to Theorem C, we can construct a (local) Riemannian metric around p such that their Ricci eigenvalues are of the form ϱ i = e 2ϕ λ i, i = 1, 2, 3 where λ 1 > λ 2 > λ 3 are nonzero constants. Denote by g such a local metric. Choose a Ricci adapted orthonormal moving frame E 1, E 2, E 3 in a neighborhood of p. Then we get ϱ ij = ϱ i δ ij = ϱ j δ ij = e 2ϕ λ i δ ij = e 2ϕ λ j δ ij for i, j = 1, 2, 3, and the expression (1) for curvature components is reduced to

R ijkl = e2ϕ n 2 (λ j(δ ik δ jl δ il δ jk ) + λ i (δ jl δ ik δ jk δ il ) + e2ϕ (λ 1 + λ 2 + λ 3 ) (δ il δ jk δ ik δ jl ), (n 1)(n 2) i, j, k, l = 1, 2, 3. In particular, we get R ijkl (p) = 1 n 2 (λ j(δ ik δ jl δ il δ jk ) + λ i (δ jl δ ik δ jk δ il ) + (λ 1 + λ 2 + λ 3 ) (n 1)(n 2) (δ ilδ jk δ ik δ jl ), i.e., R = exp(2ϕ)r p.

Now, the assumption of Lemma is satisfied and hence Proposition 2 can be applied. Thus, the corresponding metric g is (1,3)-curvature homogeneous and not (0,4)-curvature homogeneous, in general. Let us call generic those Riemannian manifolds for which the Ricci eigenvalues are distinct at all points. Due to Proposition 2, we see easily that all generic 3-dimenisonal (1,3)-curvature homogeneous Riemannian manifolds are constructed in the way described above. Hence we get the following

Theorem 1 In dimension 3, all generic real analytic (1,3)-curvature homogeneous spaces are locally parametrized, up to a local isometry, by one arbitrary real analytic function of 3 variables and three arbitrary real analytic functions of 2 variables. Hence we see that the moduli spaces of (1,3)-curvature homogeneous spaces form a one dimension bigger family than those of the (0,4)-curvature homogeneous spaces.