Hyper-parahermitian manifolds with torsion

Ivanov, Stefan; Tsanov, Vasil; Zamkovoy, Simeon

doi:10.1016/j.geomphys.2005.04.012

Cited by 10 publications

(8 citation statements)

References 25 publications

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“…Finally, given the almost para-quaternionic structure (I, J, K) induced by the Born geometry, the Born connection is also very relevant to the geometry of para-quaternionic manifolds. Indeed, similar connections have been explored in this context in [67] and a special class of such connections that have non-vanishing torsion related to integrability of the almost para-quaternionic structure via Nijenhus tensors have been explored [70]. Making the relationship with the Born connection precise is an interesting question for future research.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

A Unique Connection for Born Geometry

Freidel

Rudolph

Svoboda

2019

Commun. Math. Phys.

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It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d, d) pairing η and an O(2d) generalized metric H. More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing ω. The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure (η, ω, H) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

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Section: Conclusion and Discussionmentioning

confidence: 99%

A Unique Connection for Born Geometry

Freidel

Rudolph

Svoboda

2019

Commun. Math. Phys.

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“…On the another hand, J 1 , J 2 , J 3 being integrable, from Theorem 4.1 we deduce that the functions a, b, c also satisfy the conditions (18). The conclusion follows now easily since a, b, c, α, β , 1 a , b a(a+tb) , c a(a−tc) must be differentiable functions satisfying (18) and (26).…”

Section: The Study Of Integrabilitymentioning

confidence: 61%

“…A procedure to construct para-hyperhermitian structures on R 4n with complete and not necessarily flat associated metrics is given in [1]. Also, some examples of integrable almost para-hyperhermitian structures which admit compatible linear connections with totally skew symmetric torsion are given in [18]. Recently, in [15], a natural para-hyperhermitian structure was constructed on the tangent bundle of an almost para-hermitian manifold and on the circle bundle over a manifold with a mixed 3-structure.…”

Section: Introductionmentioning

confidence: 99%

Para-hyperhermitian structures on tangent bundles

Vîlcu¹

2011

Proc. Estonian Acad. Sci.

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In this paper we construct a family of almost para-hyperhermitian structures on the tangent bundle of an almost parahermitian manifold and study its integrability. Also, the necessary and sufficient conditions are provided for these structures to become para-hyper-Kähler.Key words: para-hyperhermitian structure, tangent bundle, paracomplex space form. PRELIMINARIESAn almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P = ±Id, such thatwhere Id is the identity tensor field of type (1,1) on M.An almost para-hermitian structure on a differentiable manifold M is a pair (P, g), where P is an almost product structure on M and g is a semi-Riemannian metric on M satisfyingIn this case, (M, P, g) is said to be an almost para-hermitian manifold. It is easy to see that the dimension of M is even. Moreover, if ∇P = 0, then (M, P, g) is said to be a para-Kähler manifold.An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such thatAn almost para-hypercomplex structure on a smooth manifold M is a triple H = (J α ) α=1,3 , where J 1 is an almost complex structure on M and J 2 , J 3 are almost product structures on M, satisfyingIn this case (M, H) is said to be an almost para-hypercomplex manifold.A semi-Riemannian metric g on (M, H) is said to be compatible or adapted to the almost parahypercomplex structure H = (J α ) α=1,3 if it satisfies Remark 2.3. If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields J 1 , J 2 , J 3 on T M by the equalities:It is easy to see that J 1 is an almost complex structure and J 2 , J 3 are almost product structures. We also have the following result (see [15]).

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“…Естественным обобщением пара-гиперкэлеровой структуры является пара-гиперкэлерова структура с кручением (ПГКК-структура), она была опреде-лена и изучена в [84]. Эта структура определяется как псевдориманова мет-рика g вместе с кососимметрической пара-гиперкомплексной структурой (J, K) и связностью ∇, которая сохраняет g, J, K и имеет кососимметрическое кру-чение T .…”

Section: специальные пара-кэлеровы многообразия Ttunclassified