Riemannian manifolds whose skewd-symmetric curvature operator has constant eigenvalues

Gilkey, Peter Β.; Leahy, John; Sadofsky, Hal

doi:10.1512/iumj.1999.48.1699

Cited by 24 publications

(39 citation statements)

References 3 publications

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“…timelike) Jordan Ivanov-Petrova for every point P of M ; again, the Jordan normal form of J R is allowed to very with the point P of M . These manifolds have been classified in the Riemannian setting if m = 3, 7 [8,11,16], in the Lorentzian setting if m ≥ 11 and if {m, m + 1} are not powers of 2 [14,15], and in the higher signature setting if q ≥ 11, if p ≤ q−6 4 , if {q, q + 1, ..., q + p} does not contain a power of 2, and if R(π) is not nilpotent [19]. We refer to [9] for further details concerning spacelike and timelike Jordan Ivanov-Petrova manifolds.…”

Section: 4mentioning

confidence: 99%

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The spectral geometry of the Weyl conformal tensor

Blažić

Gilkey

Nikčević³

et al. 2005

PDEs, Submanifolds and Affine Differential Geometry

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Abstract. We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.

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Section: 4mentioning

confidence: 99%

“…Suppose first m ≥ 5 and m = 7. We apply results of [8,11] to see that any Riemannian Ivanov-Petrova algebraic curvature tensor in these dimensions has rank 2. Such tensors are classified.…”

Section: Conformally Jordan Ivanov-petrova Manifoldsmentioning

confidence: 99%

The spectral geometry of the Weyl conformal tensor

Blažić

Gilkey

Nikčević³

et al. 2005

PDEs, Submanifolds and Affine Differential Geometry

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“…Examples of hypersurfaces that fulfill the condition 1) of Theorem 2 are IP-hypersurfaces. An IP-hypersurface in R n+1 is a hypersurface in R n+1 such that its induced metric is an IP-metric [2], [1]. The IP-metric is a warped product metric of the form ds…”

Section: Remarkmentioning

confidence: 99%

A characterization of n-dimensional hypersurfaces in Rⁿ⁺¹with commuting curvature operators

Tsankov

2005

PDEs, Submanifolds and Affine Differential Geometry

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Abstract. Let M n be a hypersurface in R n+1 . We prove that two classical Jacobi curvature operators J x and J y commute on M n , n > 2, for all orthonormal pairs (x, y) and for all points p ∈ M if and only if M n is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation (K x,y • K z,u )(u) = (K z,u • K x,y )(u), where K x,y (u) = R(x, y, u), for all orthonormal tangent vectors x, y, z, w and for all points p ∈ M .

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“…We say that a 4 tensor R ∈ ⊗ 4 V is an algebraic curvature tensor if R satisfies the symmetries given in equations (2)(3)(4). Note that we do not impose the second Bianchi identity; algebraic curvature tensors measure second order phenomena.…”

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confidence: 99%

“…Subsequently, Ivanov and Petrova [10] classified the spacelike Jordan IP metrics in the Riemannian setting for m = 4; for this reason the notation 'IP' has been used by later authors. The classification in [10] was later extended by P. Gilkey, J. Leahy, and H. Sadofsky [5] and by Gilkey [2] to the cases m ≥ 5 and m = 7. We refer to Gilkey and Semmelman [6] for some partial results if m = 7.…”

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confidence: 99%

Complex IP pseudo-Riemannian algebraic curvature tensors

Gilkey

Ivanova

2002

PDEs, Submanifolds and Affine Differential Geometry

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1. Introduction. The Riemann curvature tensor contains a great deal of information about the geometry of the underlying pseudo-Riemannian manifold; pseudo-Riemannian geometry is to a large extent the study of this tensor and its covariant derivatives. It is often convenient to work in a purely algebraic setting. We shall say that a tensor is an algebraic curvature tensor if it satisfies the symmetries of the Riemann curvature tensor. The Riemann curvature tensor defines an algebraic curvature tensor at each point of the manifold; conversely every algebraic curvature tensor is locally geometrically realizable. Thus algebraic curvature tensors are an integral part of certain questions in differential geometry.The skew-symmetric curvature operator is a natural object of study; there are other natural operators and we refer to [4] for a survey of this area. In the present paper, we examine when the (complex) Jordan normal form of the skew-symmetric curvature operator is constant either in the real or complex settings. In Section 2, we give the basic definitions and notational conventions we shall need. We also review the basic results in the real setting. In Section 3, we discuss a natural generalization to the complex setting.

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Riemannian manifolds whose skewd-symmetric curvature operator has constant eigenvalues

Cited by 24 publications

References 3 publications

The spectral geometry of the Weyl conformal tensor

The spectral geometry of the Weyl conformal tensor

A characterization of n-dimensional hypersurfaces in Rⁿ⁺¹with commuting curvature operators

Complex IP pseudo-Riemannian algebraic curvature tensors

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Riemannian manifolds whose skewd-symmetric curvature operator has constant eigenvalues

Cited by 24 publications

References 3 publications

The spectral geometry of the Weyl conformal tensor

The spectral geometry of the Weyl conformal tensor

A characterization of n-dimensional hypersurfaces in Rn+1with commuting curvature operators

Complex IP pseudo-Riemannian algebraic curvature tensors

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Product

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A characterization of n-dimensional hypersurfaces in Rⁿ⁺¹with commuting curvature operators