Cartan Connection Research Articles

The Cartan connection: sketches for a portrait of Kentaro Yano

We are describing the historical context in which Kentaro Yano prepared his doctoral dissertation under Elie Cartan’s coordination, and how this ´ work was published in Romania, with Analele S¸ tiint¸ifice ale Universit˘at¸ii “Al. I. Cuza”. We describe some of the many encounters made possible by Elie Cartan’s extraordinary creative contributions, which lead to a series of important collaborations, some of them of lasting impact until today

Lightlike manifolds and Cartan geometries

Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold N provides a lightlike metric h with radical distribution globally spanned by a vector field Z. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold (N, h), we show a method to construct a family of ambient Lorentzian manifolds that realize (N, h) as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when (N, h) is generic.

On Generalized h-Recurrent Finsler Connection

The purpose of the present paper is to generalize the concept of recurrent Finsler connection by taking h-connection by applying h-covariant derivative of Φ (p)ij as recurrent. Such a connection will be called a generalized h-recurrent Finsler connection. The relation between curvature tensors of Cartan's connection Cτ and generalized h-recurrent Finsler connection has been established.

Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family \(\pi :{{\mathcal {X}}}\rightarrow {{\mathcal {Z}}}\) of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, is \(\pi \) an S-fibration? The cases of Picard number one were answered by Hwang and Mok. The manifold \({{\mathbb {F}}}(1, Q^5)\) is the unique rational homogeneous space of Picard number one that is not rigid under Fano deformation, and a Fano degeneration of it is constructed by Pasquier and Perrin. For higher Picard number cases, one notices that the Picard number of a rational homogeneous space G/P satisfies \(\rho (G/P)\le \mathrm{rank}(G)\). Weber and Wiśniewski proved that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)\) (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper, we show that the rational homogeneous spaces G/P with \(\rho (G/P)=\mathrm{rank}(G)-1\) are rigid under Fano deformation, provided that G is a simple algebraic group of type ADE, and G/P is not biholomorphic to \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) or \({{\mathbb {F}}}(1, 2, Q^6)\). We also show that \({{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)\) has a unique Fano degeneration, which is explicitly constructed. Furthermore, the structure of possible Fano degenerations of \({{\mathbb {F}}}(1, 2, Q^6)\) is also described explicitly. Our main result is obtained by applying the theory of Cartan connections and symbol algebras.

Teleparallel gravity as a gauge theory: Coupling to matter using the Cartan connection

We present a consistent and complete description of the coupling to matter in the Teleparallel Equivalent to General Relativity (TEGR) theory built from a Cartan connection, as we proposed in previous works. A first theorem allows us to obtain parallel transport from the Cartan connection into a proper Ehresmann connection, while a second ensures to link the TEGR-Cartan connection to the Ehresmann one-form that contains the Levi-Civita connection. This yields a coupling to matter in agreement with observations and the Equivalence Principle. As the fundamental fields proceed from the Cartan connection, if one insists on interpreting TEGR as a gauge theory of translations, such translation gauge field can be extracted from the consistent theory presented. However, this would entail a fundamental change in the structures known for gauge theory and a split between gauge field and connection is imperative. The willingness to take such a step is left to the reader.

Cartan approach to Teleparallel Equivalent to General Relativity: A review

In previous works, questioning the mathematical nature of the connection in the translations gauge theory formulation of Teleparallel Equivalent to General Relativity (TEGR) Theory led us to propose a new formulation using a Cartan connection. In this review, we summarize the presentation of that proposal and discuss it from a gauge theoretic perspective.

Riemann–Cartan gravitational and axial anomalies in electrodynamics and black holes

In this paper two examples are given of a chiral torsional anomaly Nieh–Yan (NY) topological invariant in Riemann–Cartan (RC) spacetime: in the first it is shown that a certain class of RC spacetime metrics yields a chiral torsional axial anomaly possessing a term proportional to E ⋅ B. Here E and B are respectively the electric and magnetic fields. Besides, since Cartan connection does not vanish, this example is a mixed gravitational axial anomaly. In this case the axial torsional anomaly in RC manifold is computed. Second example is that of a Kerr metric of non-stationary black hole (BH) which possesses gravitational anomaly. The metric which produces the axial anomaly keeps some similarity with Eddington–Schroedinger–Einstein–Strauss metric of unified field theories of gravity and electromagnetism. These ideas, show us the possibility of having analogous BHs models in Weyl semimetals. Moreover, electrodynamics in Cartan spacetime is discussed in this context. This is based on teleparallel metrics used by Letelier (1995 Class Quantum Grav. 12 2221) and Tod (1994 Class Quantum Grav. 11 5). NY RC gravitational anomalies are obtained for BHs Kerr metric using an orthogonal tetrad system.

Structural equations of the Cartan connection with the curvature-torsion quasi-tensor

Structural equations of the Cartan connection with the curvature-torsion quasi-tensor

Symmetries of Einstein-Weyl manifolds with boundary

Starting from a real analytic surface ℳ with a real analytic conformal Cartan connection A. Bor´owka constructed a minitwistor space of an asymptotically hyperbolic Einstein–Weyl manifold with ℳ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on ℳ can be extended to symmetries of the obtained Einstein–Weyl manifold.

Recurrence Decompositions in Finsler Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.

Local Deformable 3D Reconstruction with Cartan's Connections.

3D reconstruction of deformable objects using inter-image visual motion from monocular images has been studied under Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM). Most methods have been developed for simple deformation models, primarily isometry. They may treat a surface as a discrete set of points and draw constraints from the points only or they may use a non-parametric representation and use both points and differentials to express constraints. We propose a differential framework based on Cartan's theory of connections and moving frames. It is applicable to SfT and NRSfM, and to deformation models other than isometry. It utilises infinitesimal-level assumptions on the surface's geometry and mappings. It has the following properties. 1) It allows one to derive existing solutions in a simpler way. 2) It models SfT and NRSfM in a unified way. 3) It allows us to introduce a new skewless deformation model and solve SfT and NRSfM for it. 4) It facilitates a generic solution to SfT which does not require deformation modeling. Our framework is complete: it solves deformable 3D reconstruction for a whole class of algebraic deformation models including isometry. We compared our solutions with the state-of-the-art methods and show that ours outperform in terms of both accuracy and computation time.

C-projective geometry

We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kaehler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata conjecture for complete Kaehler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric.

Existence and Uniqueness of Chern Connection in the Klein-Grifone Approach

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise comparison between Berwald, Cartan and Chern connections is presented.

Asymptotic shear and the intrinsic conformal geometry of null-infinity

In this article, we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presence of gravitational waves can be understood to be a generalization of the tractor calculus of conformal manifolds adapted to the case of degenerate conformal metrics. It follows that the whole formalism is, by construction, manifestly conformally invariant. We first show that a choice of asymptotic shear amounts to a choice of linear differential operator of order 2 on the bundle of scales of null-infinity. We refer to these operators as Poincaré operators. We then show that Poincaré operators are in one-to-one correspondence with a particular class of tractor connections, which we call “null-normal” (they generalize the normal tractor connection of conformal geometry). The tractor curvature encodes the presence of gravitational waves, and the non-uniqueness of flat null-normal tractor connections corresponds to the “degeneracy of gravity vacua” that has been extensively discussed in the literature. This work thus brings back the investigation of the radiative phase space of gravity to the study of (Cartan) connections and associated bundles. This should allow us, in particular, to proliferate invariants of the phase space.

Mollifier smoothing of C0-Finsler structures

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of $F$ with the standard mollifier in order to construct a mollifier smoothing of $F$, which is a one parameter family of Finsler structures $F_\varepsilon$ (of class $\mathit{C}^\infty$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. We prove that when $F$ is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of $F_\varepsilon$ converges uniformly on compact subsets to the corresponding objects of $F$. As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

On a $3$D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, $1$-form) modulo gauge $\widehat{g} = {\rm e}^{2\varphi} g$, $\widehat{A} = A + {\rm d}\varphi$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(\mu\nu)} - \frac{1}{3} R g_{\mu\nu} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\rm rd}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish $$ \text{W\"unschmann}(H) \equiv 0\equiv \text{Cartan}(H). $$We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to a completely similar $7$D Cartan bundle and connection. Then magically, the (complicated) equation $\text{W\"unschmann}(H) \equiv 0$ becomes $$0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x, $$ whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take $$F(x,y,z,p):= \frac{\alpha(y)(z-xp)^2+\beta(y)(z-xp)p+\gamma(y)(z-xp) +\delta(y)p^2+\varepsilon(y)p+\zeta(y)}{\lambda(y)(z-xp)+\mu(y) p+\nu(y)}, $$ with $9$ arbitrary functions $\alpha, \dots, \nu$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0 $ passes to a certain $\boldsymbol{K}(F) \equiv 0$ which holds for any choice of $\alpha(y), \dots, \nu(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $\big[ (g, A) \big]$ in $3$D. These structures are nontrivial in the sense that ${\rm d}A \not\equiv 0$ and $\text{Cotton}([g]) \not \equiv 0$.

Teleparallel theory as a gauge theory of translations: Remarks and issues

The Teleparallel Equivalent to General Relativity (TEGR) is often presented as a gauge theory of translations, i.e., that uses \emph{only} the translation group $T_4 = (\setR^4, +)$ as its gauge group. In a previous work we argued against this \emph{translation-only} formalism on the basis of its mathematical shortcomings. We then provided an alternative proposal using a Cartan connection. Recently, a reply by some of the authors defending TEGR as a \emph{translation-only} gauge theory discussed our objections. Here, we first clarify our arguments, and give new proofs of some statements, to answer to these discussions, maintaining our first claim. We then amend one of the argument that originally led us to propose the Cartan connection in this context. This broadens the \emph{a priori} possible choices for a TEGR connection.

Квазитензор кривизны-кручения фундамен­тально-групповой связности Лаптева

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations. In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.

On conformally doubly warped product Finsler manifold

The aim of the present paper is to introduce the notion of conformally doubly warped product Finsler manifold (CDWPF). For such a Finsler manifold, the coefficients of Barthel connection and its curvature tensor are investigated. The coefficients of Cartan, Berwald, Hashiguchi and Chern (Rund) connections of CDWPF are calculated. Some special Finsler spaces are studied.

Characterization of Levi-Civita and Newton–Cartan connections in dimension 2

In dimension 2, we give a local characterization of Levi-Civita connections, improving a result by G. Thompson (1991), and of Newton–Cartan connections (connections which are limit of Levi-Civita connections).