Homotheties and topology of tangent sphere bundles

Albuquerque, Rui

doi:10.1007/s00022-014-0210-x

Cited by 7 publications

(18 citation statements)

References 12 publications

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“…The formulas for Ric found in (14) match precisely with those given in the new reference. 4 The characteristic connection on the gwistor space of a flat 4-dimensional space also has parallel torsion and is harmonic as proved earlier. The G 2 -twistor structure verifies dφ = −(dµ) 2 − 2µ ∧ α 1 .…”

Section: φ) Is Given By the Torsion T C = µ ∧Dµ And Its Holonomy Is Cmentioning

confidence: 57%

“…[9, Proposition 10.80]. In particular, in dimension 4, we are left with S 4 , P 2 (C), the real hyperbolic space H 4 and the hyperbolic Hermitian space CH 2 .…”

Section: Theorem 15mentioning

confidence: 99%

“…Now, we may deduce (SS is the base manifold and π is the projection. There is a general formula in [4]. It is well known that (S ) = ≥0 = 1 for all .…”

Section: Theorem 15mentioning

confidence: 99%

“…This space admits a characteristic contact connection (T is the same viewed as a (2 1)-tensor) ∇ = ∇ + (µ/2) ∧ dµ. Thus there is no simply-connected Riemannian manifold besides S 4 whose unit tangent sphere bundle admits a characteristic contact connection, for the existence assures the manifold is K-contact.…”

Section: Assuming the Sectional Curvature Is Constant The Contact Mamentioning

confidence: 99%

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On the characteristic connection of gwistor space

Albuquerque

2013

Open Mathematics

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We give a brief presentation of gwistor spaces, which is a new concept from G 2 geometry. Then we compute the characteristic torsion T c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature and deduce the condition under which T c is ∇ c -parallel; this allows for the classification of the G 2 structure with torsion and the characteristic holonomy according to known references. The case of an Einstein base manifold is envisaged. MSC:53C10, 53C20, 53C25, 53C28Keywords: Einstein metric • gwistor space • Characteristic torsion • G 2 structure © Versita Sp. z o.o. Gwistor spaces with parallel characteristic torsion The purposeIt has now become clear that every oriented Riemannian 4-manifold M gives rise to a G 2 -twistor space, as well as its celebrated twistor space. The former was discovered in [5,6] and we shall start here by recalling how it was obtained. Often we abbreviate the name G 2 -twistor for gwistor, as suggested in [3].Given M as before, the G 2 -twistor space of M consists of a natural G 2 structure on the S 3 -bundle over M of unit tangent vectors SM = { ∈ T M : = 1}exclusively induced by the metric = · · and orientation. We shall describe the characteristic connection ∇ c of SM in the case where M is an Einstein manifold. This guarantees the gwistor structure is cocalibrated (an equivalent condition) *

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Section: φ) Is Given By the Torsion T C = µ ∧Dµ And Its Holonomy Is Cmentioning

confidence: 57%

“…[9, Proposition 10.80]. In particular, in dimension 4, we are left with S 4 , P 2 (C), the real hyperbolic space H 4 and the hyperbolic Hermitian space CH 2 .…”

Section: Theorem 15mentioning

confidence: 99%

“…Now, we may deduce (SS is the base manifold and π is the projection. There is a general formula in [4]. It is well known that (S ) = ≥0 = 1 for all .…”

Section: Theorem 15mentioning

confidence: 99%

Section: Assuming the Sectional Curvature Is Constant The Contact Mamentioning

confidence: 99%

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On the characteristic connection of gwistor space

Albuquerque

2013

Open Mathematics

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“…We have found in [9] the conditions for natural maps to become isometries between tangent sphere bundles of different radius, including weighted Sasaki metric and conformal variation of the metric on the base manifold M when dim M ≥ 3. Notice the induced horizontal subspaces on SM are not fixed on the same conformal class on M. We do not explore here these results with the weights and radius, which are all aloud to be pullbacks of functions on M.…”

Section: Further Metric Propertiesmentioning

confidence: 99%

Riemannian Questions with a Fundamental Differential System

Albuquerque

2014

Springer Proceedings in Mathematics &Amp; Statistics

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We introduce the reader to a fundamental exterior differential system of Riemannian geometry which arises naturally with every oriented Riemannian n + 1-manifold M. Such system is related to the well-known metric almost contact structure on the unit tangent sphere bundle SM, so we endeavor to include the theory in the field of contact systems. Our EDS is already known in dimensions 2 and 3, where it was used by Ph. Griffiths in applications to mechanical problems and Lagrangian systems. It is also known in any dimension but just for flat Euclidean space. Having found the Lagrangian forms α i ∈ Ω n , 0 ≤ i ≤ n, we are led to the associated functionals F i (N) = N α i , on the set of hypersurfaces N ⊂ M, and to their Poincaré-Cartan forms. A particular functional relates to scalar curvature and thus we are confronted with an interesting new equation.

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