Civita Connection Research Articles

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

Arithmetic Levi–Cività connection

This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi–Civita connection.

Some New Characterizations of Parallel Factorable Surface in Riemannian Three Dimensional Heisenberg Group

Bu çalışmada üç boyutlu Riemannian Heisenberg grubunda grup işlemiyle elde edilen paralel faktorable yüzeylerin bazı özellikleri verildi. Daha sonra H₃ te Levi- Civita konneksiyonlarına göre paralel faktorable yüzeylerin bazı karakterizasyonları elde edildi.

Laplace, Einstein and Related Equations on D-General Warping

A new concept, namely, D-general warping $$(M=M_1\times M_2,g)$$ , is introduced by extending some geometric notions defined by Blair and Tanno. Corresponding to a result of Tanno in almost contact metric geometry, an outcome in almost Hermitian context is provided here. On $$T^*M$$ , the Riemann extension (introduced by Patterson and Walker) of the Levi–Civita connection on (M, g) is characterized. A Laplacian formula of g is obtained and the harmonicity of functions and forms on (M, g) is described. Some necessary and sufficient conditions for (M, g) to be Einstein, quasi-Einstein or $$\eta $$ -Einstein are provided. The cases when the scalar (resp. sectional) curvature is positive or negative are investigated and an example is constructed. Some properties of (M, g) for being a gradient Ricci soliton are considered. In addition, D-general warpings which are space forms (resp. of quasi-constant sectional curvature in the sense of Boju, Popescu) are studied.

(Para-) Kähler Structures on $$\rho $$ ρ -Commutative Algebras

Recently, geometrics objects such as $$\rho $$ -forms, metrics, Levi–Civita connection, differential calculus were introduced on $$\rho $$ -commutative algebras. In this paper, we intend to extend another geometric objects such as almost complex structure, almost para-complex structure, Hermitian metric, para-Hermitian metric, Kahler and para-Kahler forms, on it and introduce new concepts which we will call Kahler $$\rho $$ -commutative algebras and para-Kahler $$\rho $$ -commutative algebras.

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature.

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant -sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and of the submanifold (associated with the dual connections) satisfy , i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

Invariant and anti-invariant submanifolds of special quasi-sasakian manifolds

The present paper deals with the study of Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds of SQ-Sasakian manifolds with respect to Levi–Civita connection as well as semisymmetric metric connection. It is obtained that these two classes are equivalent with a certain condition. In addition, invariant and anti-invariant submanifolds of SQ-Sasakian manifolds with respect to Levi–Civita connection as well as semisymmetric metric connection whose metrics are Ricci solitons are studied.

Structure of the Main Tensor of Conformally Connected Torsion Free Space. Conformal Connections on Hypersurfaces of Projective Space

We give a definition of a conformally connected space with an angular metric of an arbitrary signature. We present basic formulas and classes of such spaces. We obtain the decomposition of the main tensor of a conformally connected torsion free space into irreducible gauge invariant summands and prove that all affine connections obtained from the Levi–Civita connection via the normalization transformation have the same Weyl conformal tensor. We describe all conformal torsion free connections on hypersurfaces of a projective space and give some examples. We construct a global conformal connection on a hyperquadric of the projective space.

The TPS Direct Transport: A New Method for Transporting Deformations in the Size-and-Shape Space

Modern shape analysis allows the fine comparison of shape changes occurring between different objects. Very often the classic machineries of Generalized Procrustes Analysis and Principal Component Analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks. However, if size and shape data are structured in different groups thus constituting different morphological trajectories, a data centering is needed if one wants to compare solely the deformation representing the trajectories. To do that, inter-individual variation must be filtered out. This maneuver is rarely applied in studies using simulated or real data. A geometrical procedure named Parallel Transport, that can be based on various connection types, is necessary to perform such kind of data centering. Usually, the Levi Civita connection is used for interpolation of curves in a Riemannian space. It can also be used to transport a deformation. We demonstrate that this procedure does not preserve some important characters of the deformation, even in the affine case. We propose a novel procedure called `TPS Direct Transport' which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools. We recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study.

On φ-pseudo symmetric LP-Sasakian manifolds with respect to quarter-symmetric non-metric connections

The object of the present paper is to study φ-pseudo symmetric and φ-pseudo Ricci symmetric LP-Sasakian manifolds with respect to Levi–Civita connections and quarter-symmetric non-metric connections. We obtain a necessary and sufficient condition for a φ-pseudo symmetric LP-Sasakian manifold with respect to a quarter symmetric non-metric connection to be φ-pseudo symmetric LP-Sasakian manifold with respect to a Levi–Civita connection.

Real Hypersurfaces in Complex Two-plane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection

AbstractThere are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2). Among them, Suh classified Hopf hypersurfaces in G2(ℂm+2) with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster (GTW) Reeb parallel Ricci tensor for Hopf hypersurfaces in G2(ℂm+2). Next, we give a complete classification of Hopf hypersurfaces in G2(ℂm+2) with GTW Reeb parallel Ricci tensor.

A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of their Generalized Tanaka–Webster Lie Derivative

AbstractOn a real hypersurface M in a non-flat complex space form there exist the Levi–Civita and the k-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.

Pathways to relativistic curved momentum spaces: de Sitter case study

Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies, we assume that the metric of momentum space determines the condition of on-shellness while the momentum space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compare to the one most used in the recent literature. We find that the two prescriptions lead to the same picture of the so-called [Formula: see text]-momentum space, with de Sitter (dS) metric and [Formula: see text]-Poincaré connection. We then show that in the case of “proper dS momentum space”, with the dS metric and its Levi–Civita connection, the two prescriptions are inequivalent. Our novel prescription leads to a picture of proper dS momentum space which is DSR-relativistic and is characterized by a commutative law of composition of momenta, a possibility for which no explicit curved momentum space picture had been previously found. This momentum space can serve as laboratory for the exploration of the properties of DSR-relativistic theories which are not connected to group-manifold momentum spaces and Hopf algebras, and is a natural test case for the study of momentum spaces with commutative, and yet deformed, laws of composition of momenta.

Projective compactness and conformal boundaries

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $\alpha$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on $M$ which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on $M$, one obtains a projective structure on $\overline{M}$, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface $\partial M$. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.

Schmidt boundaries of foliated space-times

For every -dimensional foliated Lorentzian manifold , where is a codimension q space-like foliation, we build its Q-completion and Q-boundary . These are analogs, within transverse Lorentzian geometry of foliated manifolds, to the b-completion and b-boundary (due to (Schmidt 1971 Gen. Relativ. Gravit. 1 269–80)). The bundle morphism (mapping the -component of the Levi–Civita connection 1-form of into the unique torsion-free adapted connection on the bundle of Lorentzian transverse orthonormal frames) is shown to induce a surjective continuous map of the adapted boundary () of onto its Q-boundary. Map is used to characterize as the set of end points , in the topology of , of all Q-incomplete curves . As an application we determine a class of b-boundary points, where , g is Schwartzschildʼs metric, and is the codimension two foliation tangent to the Killing vector fields and .

On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in 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 a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

On anti-Hermitian metric connections

It is a remarkable fact that anti-Kähler and its twin metrics share the same Levi–Civita connection. Such torsion-free metric connection also emphasizes the importance of anti-Hermitian metric connections with torsion in the study of anti-Hermitian geometry. With the objective of defining new types of anti-Hermitian metric connections, in the present paper we consider classes of anti-Hermitian manifolds associated with these connections.

A classification of locally homogeneous affine connections on compact surfaces

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi–Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane. This refines previous results by Opozda.

Derivatives of Siegel modular forms and modular connections

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi–Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane. Then we show that on a Siegel upper half plane there exists at most one holomorphic $${{\rm Sp}(2g, {\mathbb {Z}})}$$ -modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.

Rate formulations in nonlinear continuum mechanics

In solving problems of geometrically nonlinear structural mechanics, a prominent role is played by formulation of rate equilibrium conditions. In the computational machinery, the evaluation of the stiffness operator provides the trial incremental displacement field as fixed point of an iterative algorithm. The issue is investigated by a new geometric approach to continuum mechanics. Kinematics is described by the motion along a trajectory manifold embedded in the affine four-dimensional space-time. Variational conditions of equilibrium and rate equilibrium are formulated in terms of natural time rates of stress and stretching. The rate elastostatic problem is formulated in the full nonlinear context by adopting a newly contributed rate-elastic constitutive model. The geometric stiffness and forcing operators are expressed in terms of an arbitrary linear spatial connection. It is shown that the adoption of a Levi- Civita connection provides a linear expression of the geometric stiffness involving a curvature term. For bodies in motion in the flat Euclid space with parallel transport by translation, a symmetric expression of the geometric stiffness is obtained, thus extending the standard formula to bodies of any dimensionality.