Loring W. Tu · Princeton University Press eBooks · 2020
This chapter discusses connections on a principal bundle. Throughout the chapter, <italic>G</italic> will be a Lie group with Lie algebra <italic>g</italic>. One possible definition of a connection on a principal <italic>G</italic>-bundle <italic>P</italic> is a <italic>C∞</italic> right-invariant horizontal distribution on <italic>P</italic>. Equivalently, a connection on <italic>P</italic> can be given by a right-equivariant <italic>g</italic>-valued 1-form on <italic>P</italic> that is the identity on vertical vectors. The chapter shows the equivalence of these two definitions of a connection. A connection is one of the most basic notions of differential geometry. It is essentially a way of differentiating sections. From a connection, the notions of curvature and geodesics follow.
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Loring W. Tu · Princeton University Press eBooks · 2020
This chapter examines curvature on a principal bundle. The curvature of a connection on a principal <italic>G</italic>-bundle is a <italic>g</italic>-valued 2-form that measures, in some sense, the deviation of the connection from the Maurer-Cartan connection on a product bundle. The Maurer-Cartan form Θ on a Lie group <italic>G</italic> satisfies the Maurer-Cartan equation. Let <italic>M</italic> be a smooth manifold. The chapter then pulls the Maurer-Cartan equation back and uses Proposition 14.3 to get the Maurer-Cartan connection. It also considers the second structural equation; the first structural equation is discussed in a previous chapter. Finally, the chapter derives some properties of the curvature form.
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Princeton University Press eBooks · 2020
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Princeton University Press eBooks · 2020
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Ronald O. Fulp · Differential Geometry and its Applications · 1994
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J. Negro, M. A. del Olmo · Journal of Mathematical Physics · 1992
In this paper the concept of bundle realization of a Lie group on an Abelian principal bundle is defined. This definition is based on the theory of locally operating realizations of Lie groups. Afterward the bundle realizations are studied and characterized into pseudoequivalence classes. This theory is applied to a systematic study and classification of invariant connections under a Lie group. In particular, some examples of gauge invariant potentials under some subgroups of the Poincaré group are worked out.
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Billaud-Friess, Marie, Antonio Falcó, Anthony Nouy · arXiv (Cornell University) · 2017
In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.
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Sebastian Brezina, Eugenia Boffo, Martin Krššák · ArXiv.org · 2025
We examine whether the Teleparallel Equivalent of General Relativity (TEGR) can be formulated as a gauge theory in the language of connections on principal bundles. We argue in favor of using either the affine bundle with the Poincaré group or, equivalently, the orthonormal frame bundle with the Lorentz group as the structure group. Following the framework of Trautman--where gauge symmetries are determined using the absolute elements--we set to identify the absolute elements and gauge symmetries of TEGR. The problem of a non-dynamical teleparallel connection raises the question of whether it should be treated as an absolute element. If so, the gauge group of TEGR is potentially some undetermined subgroup of the diffeomorphism group. On the other hand, if the connection is allowed to be non-dynamical but the only absolute element is taken to be the canonical 1-form of the frame bundle, we recover the whole diffeomorphism group as the gauge group of TEGR.
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Dale Husemöller · Graduate texts in mathematics · 1994
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Indranil Biswas · ArXiv.org · 2007
In \cite{Bi2} (Canad. Jour. Math. Vol. 58) we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in \cite{Bi2} that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalize to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.
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Andrea Nicole Young, Karen Uhlenbeck Supervisor, Dan Knopf et al. · 2013
May 2008Dedicated to my parents who always knew I could do it.Acknowledgments I would like to thank my advisor Karen Uhlenbeck for her generosity and support throughout my graduate school career. She has offered me much guidance, both professionally and personally, and I have been truly honored to work with her these past few years. I would also like to thank Dan Knopf for all that he has taught me. He too has been immensely kind and giving with his time. Thank you as well to my committee: Alan Reid, Panagiotis Souganidis, and Sonia Paban. My parents have always provided me with unconditional love and support in pursuing my goals. They have given me the wonderful gift of knowing that I am always cared for. Without my best friend Jason, I would not even have considered moving to Texas–I guess we will once again have a long-distance friendship. Girls Girls Girls and my other friends in the improv community have given me a great creative outlet and have provided me a real family here. Finally, thanks to Eric for his love and for finally realizing that I don’t actually study grouchy octopi. v Modified Ricci flow on a Principal Bundle
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Janet Talvacchia · Scholarly Commons (University of Pennsylvania) · 1989
This thesis considers the problem of prescribing curvature for connections on principal bundles. As is well known, a connection on a principal bundle uniquely determines the curvature of that bundle. A natural question to ask is, given a candidate for a curvature form, can it be realized as the curvature of a connection on the bundle? If the dimension of the base manifold is three or greater, the Bianchi identity provides an obvious restriction that the curvature form must satisfy in order to qualify as a candidate. In this thesis, we show that when the structure group of the principal bundle is semi-simple there are more non-linear identities that arise as necessary conditions on the curvature form (the number of new identities is equal to the rank of the structure group). These identities are shown to be sufficient for a real analytic curvature form to be realized locally as the curvature of a connection when the structure group of the principal bundle is SL(2,R) or SL(3,R). The method used to solve this problem is to recast it as a non-linear system of first order partial differential equations and examine the local solvability of this system. The Cartan-Kahler theorem is used to show that polynomial infinitesimal solutions can be extended to analytic series solutions.
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Tsuneyuki Yamamoto, Minoru Wakita · Journal of Periodontal Research · 1992
The bundling of principal fibers was investigated in tangential sections through the tooth-related portion in developing rat molars by light and electron microscopy. When root dentin calcification began, cross sections of principal fibers emerged as fibril aggregates in the narrow intercellular spaces in a densely packed population of periodontal ligament cells. Subsequently, these cells changed shape and location to widen the intercellular spaces. The fibril aggregates became thicker in these spaces. With root development, the collagen fibrils formed loosely aggregated bundles and the periodontal ligament cells extended cell processes between the bundles. The cell processes usually contained microfilaments suggestive of actin filaments, and as the cell processes extended and came in close apposition, they formed delimited compartments. These compartments appeared to be a sheath-like structure, and the loose fibril bundles developed into tight fibril bundles in the compartments. Finally the principal fibers consisted of many tight fibril bundles, which were partially or entirely surrounded by cell processes and cell bodies. The findings suggest that the sheath-like, cellular compartments cause the tight bundling of the principal fibers.
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R. Fioresi, Emanuele Latini, Paolo Aschieri et al. · Proceedings of Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2021) · 2022
We give a sheaf theoretic approach to the notion of quantum principal bundle over projective bases and its first order differential calculus.
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M. de León, Abel Méndez · arXiv (Cornell University) · 2015
Using a model for the bundle $\hat{\mathcal F}^2M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal F}^2M$ of holonomic second order frames of $M$, we introduce in $\hat{\mathcal F}^2M$ a principal bundle structure over ${\mathcal F}^2M$, the structure group being the additive group $A_2(n)$ of skew-symmetric bilinear maps from ${\Bbb R}^2 \times {\Bbb R}^n$ into ${\Bbb R}^n$. The composition of the projection of that structure with the existing projection of the bundle $\tilde{\mathcal F}^2M$ of non-holonomic second order frames of $M$ over $\hat{\mathcal F}^2M$ provides a principal bundle structure in $\tilde{\mathcal F}^2M$ over ${\mathcal F}^2M$. These results close an existing gap in the theory of second order frame bundles.
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CANARUTTO, DANIEL · Florence Research (University of Florence) · 1986
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Indranil Biswas, Tomás L. Gómez · Annals of Global Analysis and Geometry · 2007
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Indranil Biswas · 2008
Abstract. Let M be a C ∞ manifold and G a Lie a group. Let EG be a C ∞ princi-pal G-bundle over M. There is a fiber bundle C(EG) over M whose smooth sections correspond to the connections on EG. The pull back of EG to C(EG) has a tautological connection. We investigate the curvature of this tautological connection.
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Indranil Biswas · Canadian Journal of Mathematics · 2006
Abstract The aim here is to define connections on a parabolic principal bundle. Some applications are given.
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Frédéric Hélein · Annales de la faculté des sciences de Toulouse Mathématiques · 2025
In this paper we present original variational formulations of Yang–Mills, Einstein’s gravitation and Kaluza–Klein theories, where, in the spirit of General Relativity, the principal bundle structure over the space-time is not fixed a priori but is dynamical. In the Yang–Mills case only a topological fibration is given a priori. In the gravity and the Kaluza–Klein theories no fibration is assumed: any critical point of the action functional defines a foliation of the manifold and the leaves make up the space-time. The latter is naturally equipped with a pseudo-Riemannian metric and, under some hypotheses, this foliation is actually a fibration. In all cases the apparition of a (at least local) principal bundle structure and a connection follows from the dynamics. Moreover the metric and the connection thus constructed are solutions of the Yang–Mills, the Einstein–Cartan or the Yang–Mills–Einstein equations, depending on the model. A crucial point is that we face the difficulty that some Lagrange multiplier fields (which are responsible for the foliation, the principal bundle structure and the connection) create unwanted terms in the equations. This difficulty is overcome by the observation that, if the structure group is compact, these terms miraculously cancel.
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