A note on the structure of algebraic curvature tensors

Díaz-Ramos, José Carlos; Garcı́a-Rı́o, Eduardo

doi:10.1016/j.laa.2003.12.044

Cited by 12 publications

(20 citation statements)

References 6 publications

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“…Also, recall that the A-sectional curvature of a non-degenerate plane pZh{x, y}i is given by Further, note that the space of all algebraic curvature tensors is spanned by the A f s above (Gilkey 2001;Fiedler 2003;Díaz-Ramos & García-Río 2004).…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

2006

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Section: Algebraic Preliminariesmentioning

confidence: 99%

Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

2006

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“…If R is an algebraic curvature tensor on a two-dimensional vector space, then it is easy to see [5] that R = R ϕ for some symmetric ϕ. In addition, by replacing ϕ with −ϕ, such a ϕ can be chosen to have signature (0, 2) or (1,1).…”

Section: 1mentioning

confidence: 99%

“…Thus, M is the model of a symmetric space. (5) For generic choices of f , M f is not locally homogeneous.…”

Section: 2mentioning

confidence: 99%

On the structure group of a decomposable model space

2013

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We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature tensors on a decomposable model space must permute the subspaces V i on which they are defined. For such an algebraic curvature tensor, we show that if the vector space V is a direct sum of subspaces V 1 and V 2 , the corresponding structure group decomposes as well if V 1 and V 2 are invariant of the action of the structure group on V . We determine the freedom one has in permuting these subspaces, and show these subspaces are invariant if dim V 1 = dim V 2 or if the corresponding symmetric forms defined on those subspaces have different (but not reversed) signatures, so that in this situation, only the trivial permutation is allowable. We exhibit a model space that realizes the full permutation group, and, with exception to the balanced signature case, show the corresponding structure group is isomorphic to the wreath product of the structure group of a given symmetric bilinear form by the symmetric group. Using these results, we conclude that the structure group of any member of this family is isomorphic to a direct product of wreath products of pseudo-orthogonal groups by certain subgroups of the symmetric group. Finally, we apply our results to two families of manifolds to generate new isometry invariants that are not of Weyl type.

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“…Using (13) we transformed (12) into (14) by means of Ricci [8] in [9, curvterms.nb]. Finally, we see by a comparison with formulas in [3,Sec.4.3] that the last line of (14) is equal to 1 2 y * t (θ ⊗ F ) κλµν , where t is the tableau (5). Remark : In [3] we showed that the symmetry operator ζ −1 produces tensors U λµν of a (2 1)symmetry class which admits the index commutation symmetry U λνµ = −U λµν .…”

Section: Proof Of Theorem 110mentioning

confidence: 99%

“…The relation (9) means that the right ideal r of the symmetry class of τ λ τ [µ;ν] possesses a decomposition r = r 1 ⊕ r 2 into 2 minimal right ideals which belong to the partitions (2 1), (1 3 ) ⊢ 3. Now we see that information about the right ideal r of a symmetry class for F κ τ λ τ [µ;ν] can be gained from the Littewood-Richardson products [ Theorem 4.4 tells us that the symmetry class A(V ) is defined by the right ideal r = y * t · K[S 4 ] generated by the Young symmetrizer of the tableau (5). This right ideal r is minimal and belongs to (2 2 ) ⊢ 4 because y t ∈ a (2 2 ) .…”

mentioning

confidence: 99%