By the connection graph we mean an underlying weighted graph with a connection which associates edge set with an orthogonal group. This paper centers its investigation on the connection heat kernels on connection lattices and connection discrete torus. For one dimensional connection lattice, we derive the connection heat kernel expression by doing the Taylor expansion on the exponential function involving normalized connection Laplacian. We introduce a novel connection called product connection and prove that the connection heat kernel on arbitrary high dimensional lattice with product connection equals the Kronecker sum of one dimensional connection lattices' connection heat kernels. Furthermore, if the connection graph is consistent, we substantiate the interrelation between its connection heat kernel and its underlying graph's heat kernel. We define a connection called quotient connection such that discrete torus with quotient connection can be considered as a quotient graph of connection lattice, whose connection heat kernel is demonstrated to be the sum of connection lattices' connection heat kernels. In addition, we derive an alternative expression of connection heat kernel o
We study the perturbative path integral of Chern-Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum $\mathcal{M}' \subset \mathcal{M}$ of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on $\mathcal{M'}$ - the "global partition function" - whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples $(A,A',g)$ consisting of (1) a "kinetic" flat connection $A$ around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection $A'$, (3) a metric $g$. This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).
Adopting the pullback formalism, a new linear connection in Finsler geometry has been introduced and investigated. Such connection unifies all formerly known Finsler connections and some other connections not introduced so far. Also, our connection is a Finslerian version of the Tripathi connection introduced in Riemannian geometry. The existence and uniqueness of such connection is proved intrinsically. An explicit intrinsic expression relating this connection to Cartan connection is obtained. Some generalized Finsler connections are constructed from Tripathi Finsler connection, by applying the P1-process and C-process introduced by Matsumoto. Finally, under certain conditions, many special Finsler connections are given.
Building upon previous works characterizing GRW space-times using concircular and torse-forming vectors, this paper investigates a Lorentzian manifold equipped with a concircularly semi-symmetric metric connection. We demonstrate that such a manifold reduces to a GRW space-time under specific conditions: when the generator of the observed connection is a unit timelike vector. Also, in that case, the mentioned connection becomes a semi-symmetric metric $P$-connection. The non-zero nature of the three curvature tensors and their corresponding Ricci tensors motivates an exploration of manifold symmetries. In this way, we derive necessary and sufficient conditions for the manifold to be Einstein and we prove that a perfect fluid space-time with a semi-symmetric metric $P$-connection is Ricci pseudo-symmetric manifold of constant type. Furthermore, we show that if this space-time satisfies the Einstein's field equations without the cosmological constant, the strong energy condition is violated.
Role classification involves grouping hosts into related roles. It exposes the logical structure of a network, simplifies network management tasks such as policy checking and network segmentation, and can be used to improve the accuracy of network monitoring and analysis algorithms such as intrusion detection. This paper defines the role classification problem and introduces two practical algorithms that group hosts based on observed connection patterns while dealing with changes in these patterns over time. The algorithms have been implemented in a commercial network monitoring and analysis product for enterprise networks. Results from grouping two enterprise networks show that the number of groups identified by our algorithms can be two orders of magnitude smaller than the number of hosts and that the way our algorithms group hosts highly reflects the logical structure of the networks.
In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthogonal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserving the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenbock connections, unimodular structure, similarly appear as frame bundle reductions, with preserving connections. To each subgroup H of the linear group GL correspond reduced structures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction.
We introduce a canonical affine connection on the contact manifold $(Q,ξ)$, which is associated to each contact triad $(Q,λ,J)$ where $λ$ is a contact form and $J:ξ\to ξ$ is an endomorphism with $J^2 = -id$ compatible to $dλ$. We call it the \emph{contact triad connection} of $(Q,λ,J)$ and prove its existence and uniqueness. The connection is canonical in that the pull-back connection $φ^* abla$ of a triad connection $ abla$ becomes the triad connection of the pull-back triad $(Q, φ^*λ, φ^*J)$ for any diffeomorphism $φ:Q \to Q$ satisfying $φ^*λ= λ$ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric $$ g_{(λ,J)} = dλ(\cdot, J\cdot) + λ\otimes λ$$ and $J$ regarded as an endomorphism on $TQ = \mathbb R\{X_λ\}\oplus ξ$, and is characterized by its torsion properties and the requirement that the contact form $λ$ be holomorphic in the $CR$-sense. In particular, the connection restricts to a Hermitian connection $ abla^π$ on the Hermitian vector bundle $(ξ,J,g_ξ)$ with $g_ξ= dλ(\cdot, J\cdot)|_ξ$, which we call the \emph{contact Hermitian connection} of $(ξ,J,g_ξ)$. These connections greatly simplify tensorial calculations in the sequels \cite{oh-w
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen's inequalities for non integrable distributions of real space forms endowed with a semi-symmetric metric connection and a semi-symmetric non-metric connection. We give some examples of non integrable distributions in a Riemannian manifold with affine connections. We find some new examples of Einstein distributions and distributions with constant scalar curvature.
The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type $(1,2)$, by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $N= 1$ super-Minkowski spacetime admits a natural odd connection.
The Barbero-Immirzi (BI) connection, as usually introduced out of a spin connection, is a global object though it does not transform properly as a genuine connection with respect to generic spin transformations, unless quite specific and suitable gauges are imposed. We shall here investigate whether and under which global conditions a (properly transforming and hence global) SU(2)-connection can be canonically defined in a gauge covariant way in such a way that SU(2)-connection locally agrees with the usual BI connection and can be defined on pretty general bundles (in particular triviality is not assumed). As a by-product we shall also introduce a global covariant SU(2)-connection over the whole spacetime (while for technical reasons the BI connection in the standard formulation is just introduced on a space slice) which restricts to the usual BI connection on a space slice.
An axiomatic theory of operator connections and operator means was investigated by Kubo and Ando in 1980. A connection is a binary operation for positive operators satisfying the monotonicity, the transformer inequality and the joint-continuity from above. In this paper, we show that the joint-continuity assumption can be relaxed to some conditions which are weaker than the separate-continuity. This provides an easier way for checking whether a given binary opertion is a connection. Various axiomatic characterizations of connections are obtained. We show that the concavity is an important property of a connection by showing that the monotonicity can be replaced by the concavity or the midpoint concavity. Each operator connection induces a unique scalar connection. Moreover, there is an affine order isomorphism between connections and induced connections. This gives a natural viewpoint to define any named means.
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-defined requires the cosmological constant to be quantised. Finally, in the connection setting one can deform the 4D Einstein condition in an interesting way, and we show that a
Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps determining projections of the tangent bundle onto the partial connection; this approach eliminates many of the complications arising from the presence of isotropy. A connection form taking values in the dual of the Lie algebra is smooth even at singular points of the action, while analogs of the classical algebra-valued connection form are necessarily discontinuous at such points. The curvature of a partial connection form can be defined under mild technical hypotheses; the interpretation of curvature as a measure of the lack of involutivity of the (partial) connection carries over to this general setting.
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkowskian connections which can be considered as special types of affine connections, the group of affine transformations of the affine tangent space being replaced by a suitable subgroup; and more generally, conformal and projective connections, associated to a group which is no more a subgroup of the affine group. Around 1950, Charles Ehresmann introduced connections on a fibre bundle and, when the bundle has a Lie group as structure group, connection forms on the associated principal bundle, with values in the Lie algebra of the structure group. He called Cartan connections the various types of
By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.
For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange structure. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonic nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. The metric tensor of the Lagrange space determines the symmetric part of the nonlinear connection, while the symplectic structure of the Lagrange space determines the skew-symmetric part of the nonlinear connection.
We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $ abla\colon \mathfrak{X}(M)\timesΓ(E)\toΓ(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\oplus H_ abla$, where $T^{q_E}E$ is the set of vectors tangent to the fibres of $E$. Furthermore, the curvature of the connection measures the failure of the horizontal space $H_ abla$ to be integrable. We show that linear horizontal complements to $T^{q_E}E\oplus (T^{q_E}E)^\circ$ in the Pontryagin bundle over the vector bundle $E$ can be described in the same manner via a certain class of Dorfman connections $Δ\colon Γ(TM\oplus E^*)\timesΓ(E\oplus T^*M)\toΓ(E\oplus T^*M)$. Similarly to the tangent bundle case, we find that, after the choice of a linear splitting, the standard Courant algebroid structure of $TE\oplus T^*E\to E$ can be completely described by properties of the Dorfman connection. As an application, we study splittings of $TA\oplus T^*A$ over a Lie algebroid $A$ and, following Gracia-Saz and Mehta, we compute the representations up to homotopy defined by
The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the r
Rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $δ$, the rainbow connection number is upper bounded by $3n/(δ+ 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(λ+ 1) + 3$ and $3n/(κ+ 1) + 3$ for rainbow connection number where $λ$ and $κ$, respectively, denote the edge and vertex connectivity of the graph. We show that the above bound in terms of edge connectivity is tight up-to additive constants and show that the bound in terms of vertex connectivity can be improved to $(2 + ε)n/κ+ 23/ ε^2$, for any $ε> 0$. We conjecture that rainbow connection number is upper bounded by $n/κ+ O(1)$ and show that it is true for $κ= 2$. We also show that the conjecture is true for chordal graphs and graphs of girth at least 7.
We study a homogeneous system of $d+8$ linear partial differential equations (PDEs) in $d$ variables arising from two-dimensional Conformal Field Theories (CFTs) with a $W_3$-symmetry algebra. In the CFT context, $d$ PDEs are third-order and correspond to the null-state equations, whereas the remaining 8 PDEs (five being second-order and three being first-order) correspond to the $W_3$ global Ward identities. In the case of central charge $c=2$, we construct a subspace of the space of all solutions which grow no faster than a power law. We call this subspace the space of $W_3$ conformal blocks, and we provide a basis expressed in terms of Specht polynomials associated with column-strict, rectangular Young tableaux with three columns. The dimension of this space is a Kostka number which coincides with CFT predictions, hence we conjecture that it exhausts the space of all solutions having a power law bound. Moreover, we prove that the space of $W_3$ conformal blocks is an irreducible representation of a certain diagram algebra defined from $\mathfrak{sl}_3$ webs that we call Kuperberg algebra. Finally, we prove a formula relating the $W_3$ conformal blocks at $c=2$ we constructed to