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On the frame bundle adapted to a submanifold  [PDF]
Kamil Niedzialomski
Mathematics , 2013,
Abstract: Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distribution of $O(M,N)$ and state its correspondence with the horizontal lift in $O(N)$ induced by the Levi--Civita connection on $N$. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold $M$ with deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures.
The natural operators lifting vector fields to the bundle of affinors
Jan Kurek , W odzimierz M. Mikulski
Annales UMCS, Mathematica , 2008, DOI: 10.2478/v10062-008-0009-z
Abstract: All natural operators T T(T T*) lifting vector fields X from n-dimensional manifolds M to vector fields B(X) on the bundle of affinors T*M are described.
Canonical vector valued 1-forms on higher order adapted frame bundles over category of fibered squares
Anna Bednarska
Annales UMCS, Mathematica , 2008, DOI: 10.2478/v10062-008-0003-5
Abstract: Let Y be a fibered square of dimension (m1, m2, n1, n2). Let V be a finite dimensional vector space over. We describe all 21,m2,n1,n2 - canonical V -valued 1-form Θ TPrA (Y) → V on the r-th order adapted frame bundle PrA(Y).
An Adapted Frame on Indicatrix Bundle of a Finsler Manifold and its Geometric Properties  [PDF]
H. Attarchi,M. M. Rezaii
Mathematics , 2011,
Abstract: In this paper, a frame is introduced on tangent bundle of a Finsler manifold in a manner that it makes some simplicity to study the properties of the natural foliations in tangent bundle. Moreover, we show that the indicatrix bundle of a Finsler manifold with lifted sasaki metric and natural almost complex structure on tangent bundle cannot be a sasakian manifold.
Underlying functors on fibered manifolds  [cached]
Miroslav Doupovec
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica , 2004,
Abstract: For a product preserving bundle functor on the category of fibered manifolds we describe subordinated functors and we introduce the concept of the underlying functor. We also show that there is an affine bundle structure on product preserving functors on fibered manifolds.
Resolvent, heat kernel and torsion under degeneration to fibered cusps  [PDF]
Pierre Albin,Frédéric Rochon,David Sher
Mathematics , 2014,
Abstract: Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.
A frame bundle generalization of multisymplectic field theories  [PDF]
J. K. Lawson
Physics , 1997, DOI: 10.1016/S0034-4877(00)89031-X
Abstract: This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generalized symplectic structure from the full frame bundle. The geometric structure of the vertically adapted frame bundle admits vector-valued field observables and produces vector-valued Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We show that the linear and affine multivelocity spaces and multiphase spaces for geometric field theories are associated to the vertically adapted frame bundle. In addition, the new geometry not only generalizes both the linear and the affine models of multisymplectic geometry but also resolves fundamental problems found in both multisymplectic models.
Fibered Correspondence  [PDF]
Aleks Kleyn
Mathematics , 2007,
Abstract: Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. Reduced fibered correspondence in bundle is called 2-ary fibered relation. We considered fibered equivalence and isomorphism theorem in case of fibered morphisms.
The Geometry of the Frame Bundle over Spacetime  [PDF]
Fredrik St?hl
Mathematics , 2000,
Abstract: One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metric G. We study the geometry of (LM,G) as a Riemannian manifold in the case when the connection is the Levi-Civita connection of a Lorentzian metric g on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of (LM,G) in relation to the geometry of (M,g).
A Frame Bundle Generalization of Multisymplectic Momentum Mappings  [PDF]
J. K. Lawson
Mathematics , 2001, DOI: 10.1016/S0034-4877(04)90002-X
Abstract: This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field momentum observables on $Z$ are generalized by those on $L_VY$. Furthermore, momentum observables on $L_VY$ produce conserved quantities along flows in $L_VY$. For translational and orthogonal symmetries of fields and reparametrization symmetry in mechanics, momentum is conserved, and for angular momentum in time-evolution mechanics we produce a version of the parallel axis theorem of rotational dynamics, and in special relativity, we produce the transformation of angular momentum under boosts.
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