Complex Structures on the Tangent Bundle of Riemannian Manifolds

1993 ◽  
pp. 235-251 ◽  
Author(s):  
Làszló Lempert
Keyword(s):  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Complex Structures

scholarly journals A Kähler Einstein structure on the tangent bundle of a space form

2001 ◽  
Vol 25 (3) ◽  
pp. 183-195 ◽  
Author(s):  
Vasile Oproiu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Space Form ◽  
Positive Curvature ◽  
Holomorphic Sectional Curvature ◽  
Zero Section ◽  
Constant Negative Curvature ◽  
Constant Positive Curvature

We obtain a Kähler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kähler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

scholarly journals Symplectic and Contact Geometry with Complex Manifolds

2019 ◽  
Vol 39 ◽  
pp. 119-126
Author(s):  
AKM Nazimuddin ◽  
Md Showkat Ali
Keyword(s):  
Riemannian Manifolds ◽  
Symplectic Geometry ◽  
Contact Geometry ◽  
Symplectic Manifolds ◽  
Complex Structures ◽  
Contact Manifolds ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Real Submanifold ◽  
Complex Symplectic

In this paper, we discuss about almost complex structures and complex structures on Riemannian manifolds, symplectic manifolds and contact manifolds. We have also shown a special comparison between complex symplectic geometry and complex contact geometry. Also, the existence of a complex submanifold of n-dimensional complex manifold which intersects a real submanifold GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 119-126

Complex structures on tangent bundles of Riemannian manifolds

1991 ◽  
Vol 291 (1) ◽  
pp. 409-428 ◽  
Author(s):  
Róbert Szőke
Keyword(s):  
Riemannian Manifolds ◽  
Complex Structures ◽  
Tangent Bundles

Remarks about the Kaluza-Klein metric on tangent bundle

2019 ◽  
Vol 16 (03) ◽  
pp. 1950040
Author(s):  
Murat Altunbas ◽  
Lokman Bilen ◽  
Aydin Gezer
Keyword(s):  
Tangent Bundle ◽  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex ◽  
Compatible Almost Complex Structure ◽  
Almost Kähler ◽  
Necessary And Sufficient ◽  
Curvature Tensors ◽  
Kaluza Klein

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.

scholarly journals Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

2019 ◽  
Vol 63 (1) ◽  
pp. 118-140 ◽  
Author(s):  
Danka Lučić ◽  
Enrico Pasqualetto
Keyword(s):  
Hilbert Space ◽  
Riemannian Manifold ◽  
Sobolev Space ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Radon Measure ◽  
Finsler Manifold ◽  
Carnot Groups

AbstractThe main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.

scholarly journals Adapted complex structures and geometric quantization

1999 ◽  
Vol 154 ◽  
pp. 171-183 ◽  
Author(s):  
Róbert Szőke
Keyword(s):  
Symmetric Space ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Symmetric Spaces ◽  
Complex Structure ◽  
Geometric Quantization ◽  
Riemannian Symmetric Space ◽  
Complex Structures ◽  
Rank One ◽  
Push Forward

AbstractA compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS.

scholarly journals Singular manifolds of proteomic drivers to model the evolution of inflammatory bowel disease status

2019 ◽  
Author(s):  
Ian Morilla ◽  
Thibaut Léger ◽  
Assiya Marah ◽  
Isabelle Pic ◽  
Hatem Zaag ◽  
...  
Keyword(s):  
Riemannian Manifolds ◽  
Disease Status ◽  
Graph Laplacian ◽  
Complex Structures ◽  
Geometric Framework ◽  
Topological Overlap ◽  
Bowel Diseases ◽  
Inflammatory Bowel ◽  
Different Levels ◽  
Singular Manifolds

The conditions who denotes the presence of an immune disease are often represented by interaction graphs. These informative, but complex structures are susceptible to being perturbed at different levels. The mode in which that perturbation occurs is still of utmost importance in areas such as reprogramming therapeutics. In this sense, the overall graph architecture is well characterise by module identification. Topological overlap-related measures make possible the localisation of highly specific module regulators that can perturb other nodes, potentially causing the entire system to change behaviour or collapse. We provide a geometric framework explaining such situations in the context of inflammatory bowel diseases (IBD). IBD are important chronic disorders of the gastrointestinal tract which incidence is dramatically increasing worldwide. Our approach models different IBD status as Riemannian manifolds defined by the graph Laplacian of two high throughput proteome screenings. Identifies module regulators as singularities within the manifolds (the so-called singular manifolds). And reinterprets the characteristic IBD nonlinear dynamics as compensatory responses to perturbations on those singularities. Thus, we could control the evolution of the disease status by reconfiguring particular setups of immune system to an innocuous target state.

scholarly journals Harmonic morphisms from even-dimensional hyperbolic spaces

2003 ◽  
Vol 92 (2) ◽  
pp. 246 ◽  
Author(s):  
Martin Svensson
Keyword(s):  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Hyperbolic Spaces ◽  
Harmonic Morphisms ◽  
Totally Geodesic ◽  
Complex Valued

In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

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