Geometry of the Total Space of a Tangent Bundle

1994 ◽  
pp. 106-128
Author(s):  
Radu Miron ◽  
Mihai Anastasiei
Keyword(s):  
Tangent Bundle ◽  
Total Space

scholarly journals The projective curvature of the tangent bundle with natural diagonal metric

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 401-410 ◽  
Author(s):  
Cornelia-Livia Bejan ◽  
Simona-Luiza Druţă-Romaniuc
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Space ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Necessary And Sufficient

Our study is mainly devoted to a natural diagonal metric G on the total space TMof the tangent bundle of a Riemannian manifold (M, 1). We provide the necessary and sufficient conditions under which (TM,G) is a space form, or equivalently (TM,G) is projectively Euclidean. Moreover, we classify the natural diagonal metrics G for which (TM,G) is horizontally projectively flat (resp. vertically projectively flat).

scholarly journals Complex hyperkähler structures defined by Donaldson–Thomas invariants

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Tom Bridgeland ◽  
Ian A. B. Strachan
Keyword(s):  
Complex Manifold ◽  
Geometric Structure ◽  
Tangent Bundle ◽  
Total Space ◽  
Stability Conditions ◽  
Link Type ◽  
Hyperkähler Metrics

AbstractThe notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a $$\hbox {CY}_3$$ CY 3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

scholarly journals DIRAC STRUCTURES FROM LIE INTEGRABILITY

2012 ◽  
Vol 09 (04) ◽  
pp. 1220005 ◽  
Author(s):  
MIRCEA CRASMAREANU
Keyword(s):  
Tangent Bundle ◽  
Riemannian Geometry ◽  
Fiber Bundle ◽  
Symplectic Structure ◽  
Total Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
3 Dimensional ◽  
Ehresmann Connection

We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

scholarly journals Lagrange geometry on tangent manifolds

2003 ◽  
Vol 2003 (51) ◽  
pp. 3241-3266 ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Tangent Bundle ◽  
Tensor Field ◽  
Finsler Geometry ◽  
Interesting Case ◽  
Lagrangian Function ◽  
Total Space ◽  
Lagrangian Functions ◽  
Tangent Manifold

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index-free presentation of these connections.

Golden Riemannian structures on the tangent bundle with g-natural metrics

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2543-2554
Author(s):  
E. Peyghan ◽  
F. Firuzi ◽  
U.C. De
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Riemannian Metric

Starting from the g-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g), we construct a family of the Golden Riemannian structures ? on the tangent bundle (TM,G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM,?,G) and the locally flatness of manifold (M,g).

Families over a base with a birationally nef tangent bundle

1997 ◽  
Vol 308 (2) ◽  
pp. 347-359 ◽  
Author(s):  
Sándor J. Kovács
Keyword(s):  
Tangent Bundle

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

scholarly journals Variation formulas for principal functions, II: Applications to variation for harmonic spans

2011 ◽  
Vol 204 ◽  
pp. 19-56 ◽  
Author(s):  
Sachiko Hamano ◽  
Fumio Maitani ◽  
Hiroshi Yamaguchi
Keyword(s):  
Direct Application ◽  
Total Space ◽  
Geometric Meaning ◽  
Principal Function ◽  
Variation Formula ◽  
Circular Slit ◽  
Moving Domain

AbstractA domainD⊂ Czadmits the circular slit mappingP(z) fora, b∈Dsuch thatP(z) – 1/(z–a) is regular ataandP(b) = 0. We callp(z) =log|P(z)|theLi-principal functionandα= log |P′(b)| theL1-constant, and similarly, the radial slit mappingQ(z) implies theL0-principal functionq(z) and theL0-constantβ. We calls=α–βtheharmonic spanfor (D, a, b). We show the geometric meaning ofs. Hamano showed the variation formula for theL1-constantα(t) for the moving domainD(t) in Czwitht∈B:= {t∈ C: |t| <ρ}. We show the corresponding formula for theL0-constantβ(t) forD(t) and combine these to prove that, if the total spaceD =∪t∈B(t, D(t)) is pseudoconvex inB× Cz, thens(t) is subharmonic onB. As a direct application, we have the subharmonicity of log coshd(t) onB, whered(t) is the Poincaré distance betweenaandbonD(t).

scholarly journals $$ \mathcal{N} $$ = 2 consistent truncations from wrapped M5-branes

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Davide Cassani ◽  
Grégoire Josse ◽  
Michela Petrini ◽  
Daniel Waldram
Keyword(s):  
Riemann Surface ◽  
Gauge Group ◽  
Tangent Bundle ◽  
Dimensional Manifold ◽  
Five Dimensions ◽  
Intrinsic Torsion ◽  
Consistent Truncations

Abstract We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal $$ \mathcal{N} $$ N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5×wM solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the $$ \mathcal{N} $$ N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.

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