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In this article, we first introduce new Frenet formulas by making use of the properties of connected curves. Then applying these formulas we show that some decisive properties of Bertrand partner curve can be given in terms of a Bertrand curve. More precisely, we offer differential equations of the Bertrand partner curve with respect to both Levi-Civita and normal Levi-Civita connections in terms of the Bertrand curve. We also give harmonicity conditions of the partner curve of Bertrand curve pair by the same method. We obtain some new results and finally we give an example to support our allegations. [ABSTRACT FROM AUTHOR]

Our work is study the geometry of oscillator groups; they are the only non commutative simply connected solvable Lie groups which have a biinvariant Lorentzian metric. The oscillator group has been generalized to one dimension equals 2n x 4, and several aspects of its geometry have been intensively studied, both in differential geometry and in mathematical physics. In this paper, we find geodesic curves on the oscillator group (Gλ,gα) of dimension four. [ABSTRACT FROM AUTHOR]

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an N -dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The G L (N , ℝ) symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is U (1) and a seven-dimensional model in which it is S U (2). We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory. [ABSTRACT FROM AUTHOR]

We study covariant Sobolev spaces and ∇ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted ∇ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the ∇ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection ∇ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our ∇ -differential operators. We also introduce and study the notion of a ∇ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for ∇ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience. [ABSTRACT FROM AUTHOR]

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton's equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e. finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion free connections for every minimal surface in this class. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of Levi-Civita connections for strongly σ -compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in [J. Bhowmick, D. Goswami and S. Mukhopadhyay, Levi-Civita connections for a class of spectral triples, Lett. Math. Phys. 110 (2020) 835–884] and [J. Bhowmick, D. Goswami and G. Landi, On the Koszul formula in noncommutative geometry, Rev. Math. Phys. 32(10) (2020) 2050032]. [ABSTRACT FROM AUTHOR]

Given a bicovariant differential calculus (E , d) such that the braiding map is diagonalisable in a certain sense, the bimodule of two-tensors admits a direct sum decomposition into symmetric and anti-symmetric tensors. This is used to prove the existence of a bicovariant torsionless connection on E. Following Heckenberger and Schmüdgen, we study invariant metrics and the compatibility of covariant connections with such metrics. A sufficient condition for the existence and uniqueness of bicovariant Levi-Civita connections is derived. This condition is shown to hold for cocycle deformations of classical Lie groups. [ABSTRACT FROM AUTHOR]

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric. [ABSTRACT FROM AUTHOR]

We study covariant derivatives on a class of centered bimodules ℰ over an algebra 𝒜. We begin by identifying a 𝒵 (𝒜) -submodule 𝒳 (𝒜) which can be viewed as the analogue of vector fields in this context; 𝒳 (𝒜) is proven to be a Lie algebra. Connections on ℰ are in one-to-one correspondence with covariant derivatives on 𝒳 (𝒜). We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived. [ABSTRACT FROM AUTHOR]

In this paper, we use of the geometry of a class of the nature flows to define trajectory manifolds. Trajectory connections as a generalization of the Levi-Civita connections are considered. A method for determining the geometry of the flows created by the integral curves of a vector field is presented. The method contains two steps, the first step is finding the connection by the trajectories of a vector field, and the second step is finding a trajectory metric corresponding to the deduced connection. We show that doing the first step is possible, but for some of the vector fields, the second step may not be possible. In the case of existence of a trajectory manifold a new kind of curvature which we called it "trajectory curvature scalar" appears. We calculate trajectory connections for some vector fields and by an example we show that the trajectory curvature scalar for a trajectory manifold may not be equal to the curvature scalar of it. We find trajectory connection for a vector field close to the Schwarzschild black hole. [ABSTRACT FROM AUTHOR]

In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection. [ABSTRACT FROM AUTHOR]

A differential calculus on the Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I–IV of Bhowmick et al. (A new look at Levi-Civita Connection in NCG. arXiv:1606.08142) so that Levi-Civita connection exists uniquely for any pseudo-Riemannian metric in the sense of Bhowmick et al. [2] Scalar curvature is computed for the Levi-Civita connection corresponding to the canonical bilinear metric. [ABSTRACT FROM AUTHOR]

Abstract We introduce Kähler–Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a Kähler–Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated Kähler–Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts. [ABSTRACT FROM AUTHOR]

In this paper, some characterizations of surfaces which are constructed by using the group operation in 3 Heis are given. Then, a new classification for minimal factorable surfaces in 3 Heis is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine conformally flat, locally strongly convex affine hypersurfaces with pseudo-parallel cubic form relative to the Levi-Civita connection of affine metric. Such hypersurfaces include two natural classes: all affine surfaces; all affine hypersurfaces with constant sectional curvature. In particular, the classification of 3-dimensional locally strongly convex affine hypersurfaces with pseudo-parallel cubic form is established. [ABSTRACT FROM AUTHOR]

In this paper, the object is to study a semi-symmetric non-metric connection on an LP-Sasakian manifold whose concircular curvature tensor satisfies certain curvature conditions. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study a special type of quarter-symmetric non-metric connection on a P-Sasakian manifold. It is shown that the first Bianchi identity of the curvature tensors on P-Sasakian manifolds admits a special type of quarter-symmetric non-metric connection. Among others we prove that if PSasakian manifolds admit a special type of quarter-symmetric non-metric connection, then they are Ricci-Semi-symmetric. Finally, an illustrative example is given to verify our result. [ABSTRACT FROM AUTHOR]

We make some observations about Rosenberg's Levi-Civita connections on noncommutative tori, noting the non-uniqueness of general torsion-free metric-compatible connections without prescribed connection operator for the inner *-derivations, the nontrivial curvature form of the inner *-derivations, and the validity of the Gauss-Bonnet theorem for two classes of nonconformal deformations of the flat metric on the noncommutative two-tori, including the case of noncommuting scalings along the principal directions of a two-torus. [ABSTRACT FROM AUTHOR]

On a real hypersurface M in a complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the k -th g-Tanaka–Webster connection. Associated to g-Tanaka–Webster connection we can define a differential operator of first order. We classify real hypersurfaces such that both the Lie derivative and this differential operator, either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the structure Jacobi operator of M . [ABSTRACT FROM AUTHOR]

We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Chern–Gauss–Bonnet type theorem for the noncommutative 4-sphere. [ABSTRACT FROM AUTHOR]

In this paper we try to extend geometric concepts in the context of operator valued tensors. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commutative C*-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature tensor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. [ABSTRACT FROM AUTHOR]

We construct tame differential calculi coming from toral actions on a class of C ⁎ -algebras. Relying on the existence of a unique Levi-Civita connection on such a calculus, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical tame calculus of rank two is studied. [ABSTRACT FROM AUTHOR]

In this study, Fermi derivative and according to the derivative Fermi parallelism and non-rotating frame concepts are given on any hypersurface in En+1. Initially, Fermi derivative and non-rotating concepts are analyzed on any hypersurface in E4. As an example, S3 is taken instead of the hypersurface in E4. Fermi derivative and non-rotating concepts are examined for S3. Then a correlation is found between Fermi derivative and Levi-Civita derivative of any vector field in E4. And then the concepts of Fermi derivative are generalized for n > 4 in En+1. [ABSTRACT FROM AUTHOR]

We consider two Riemannian geometries for the manifold $${\mathcal{M }(p,m\times n)}$$ of all $$m\times n$$ matrices of rank $$p$$ . The geometries are induced on $${\mathcal{M }(p,m\times n)}$$ by viewing it as the base manifold of the submersion $$\pi :(M,N)\mapsto MN^\mathrm{T}$$ , selecting an adequate Riemannian metric on the total space, and turning $$\pi $$ into a Riemannian submersion. The theory of Riemannian submersions, an important tool in Riemannian geometry, makes it possible to obtain expressions for fundamental geometric objects on $${\mathcal{M }(p,m\times n)}$$ and to formulate the Riemannian Newton methods on $${\mathcal{M }(p,m\times n)}$$ induced by these two geometries. The Riemannian Newton methods admit a stronger and more streamlined convergence analysis than the Euclidean counterpart, and the computational overhead due to the Riemannian geometric machinery is shown to be mild. Potential applications include low-rank matrix completion and other low-rank matrix approximation problems. [ABSTRACT FROM AUTHOR]

Subject of present paper is the geometry of foliation defined by closed 1-form on compact manifold of constant curvature. In the paper everywhere smoothness of manifolds, foliations and maps is a class C∞. [ABSTRACT FROM AUTHOR]

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Separately, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean R n corresponding to the affine orthogonal Lie algebra, n ≥ 3. [ABSTRACT FROM AUTHOR]

For an arbitrary subalgebra h ⊂ so( r, s) a polynomial pseudo-Riemannian metric of signature ( r + 2, s + 2) is constructed, the holonomy algebra of this metric contains h as a subalgebra. This result shows the essential distinction between the holonomy algebras of pseudo-Riemannian manifolds of index greater than or equal to 2 and the holonomy algebras of Riemannian and Lorentzian manifolds. [ABSTRACT FROM AUTHOR]

We prove that a vector field X on a compact Riemannian manifold ( M, g) with Levi-Cività connection ∇ is an infinitesimal isometry if and only if it satisfies the system of differential equations: trace( L∇) = 0, trace( L Ric) = 0, where L is the Lie derivative in the direction of X and Ric is the Ricci tensor. It follows from the second assertion that the Ricci soliton on a compact manifold M is trivial if its vector field X satisfies one of the following two conditions: trace( L Ric) ≤ 0 or trace( L Ric) ≥ 0. [ABSTRACT FROM AUTHOR]

Abstract: This Letter considers the curvature conditions of AW(k)-type null Cartan curves, and investigates null Cartan Bertrand curves. We show that null Cartan Bertrand curves are AW(k)-type curves in Minkowski 4-space. [Copyright &y& Elsevier]

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (and Riemannian real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor. [ABSTRACT FROM AUTHOR]

The article discusses an affine contact structure on an odd-dimensional smooth manifold with a Lie group of faithful affine contact transformations of maximal dimension. It says that a curvature tensor algebraic structure for the symmetric region and the torsion tensor of the connection structure exists. Furthermore, the maximal dimension of the lie group is equal to 2n² + 3n + 1.

Abstract: We present a number of conditions which are necessary for an -dimensional projective structure to include the Levi-Civita connection of some metric on . We provide an algorithm, which effectively checks whether a Levi-Civita connection is in the projective class and, which finds this connection and the metric, when it is possible. The article also provides basic information on invariants of projective structures, including the treatment via the Cartan normal projective connection. In particular we show that there are a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about . [Copyright &y& Elsevier]

Abstract: In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric. [Copyright &y& Elsevier]

We study on a Riemannian manifold (M, g) with a semi-symmetric non-metric connection. We obtain some characterizations for (M, g) satisfying some semisymmety conditions. [ABSTRACT FROM AUTHOR]

Abstract: We study Lorentzian affine hypersurfaces in with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in is given. [Copyright &y& Elsevier]

Several important design problems in signal processing for communications can be cast as optimization problems in which the objective is a function of the subspaces spanned by tall complex matrix variables with orthonormal columns. Such problems can be viewed as optimization problems on the complex Grassmann manifold, and an effective means for performing this optimization is to use a Grassmannian version of Newton's method. To facilitate the implementation of that method, we provide an explicit expression for the Grassmannian Newton direction for an arbitrary twice differentiable function. We also use an example in which the pairwise chordal Frobenius norm between subspaces is to be optimized to outline a systematic procedure for obtaining the Hessian matrix. [ABSTRACT FROM AUTHOR]

Abstract: We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the -algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that -compatible bimodule connections lead to braid operators in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert -modules and metric compatibility in both the usual form and a cotorsion form. We show that the theory works well for the quantum group with its three-dimensional calculus, finding for each point of a three-parameter space of covariant metrics a unique ‘Levi-Civita’ connection deforming the classical one and characterised by zero torsion, metric preservation and -compatibility. Allowing torsion, we find a unique connection with a classical limit that is metric preserving and -compatible and for which obeys the braid relations. It projects to a unique ‘Levi-Civita’ connection on the quantum sphere. The theory also works for finite groups, and in particular for the permutation group , where we find somewhat similar results. [ABSTRACT FROM AUTHOR]

Locally convex compact immersed hypersurfaces in the Finsler—Hadamard space with bounded T-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity of parallel hypersurfaces is proved. [ABSTRACT FROM AUTHOR]

We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion. [ABSTRACT FROM AUTHOR]

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature. [ABSTRACT FROM AUTHOR]

We study Riemannian manifolds M admitting a semi-symmetric metric connection ∇̃ such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection ∇. We prove that R·R̃ = 0 if and only if M is semisymmetric; if R̃·R = 0 or R·R̃ - R·R̃·R = 0 or M is semisymmetric and R̃·R̃ = 0, then M is conformally flat and quasi-Einstein. Here R and R̃ denote the curvature tensors of ∇ and ∇̃, respectively. [ABSTRACT FROM AUTHOR]

In this paper, it is obtained lifts from a differentiable manifold has basic tensor fields ie function, vector field, 1-form to its double tangent bundles. [ABSTRACT FROM AUTHOR]

We derive the equations of Gauss and Weingarten for a non-degenerate hypersurface of a semi-Riemannian manifold admitting a semi-symmetric metric connection, and give some corollaries of these equations. In addition, we obtain the equations of Gauss curvature and Codazzi-Mainardi for this non-degenerate hypersurface and give a relation between the Ricci and the scalar curvatures of a semi-Riemannian manifold and of its a non-degenerate hypersurface with respect to a semi-symmetric metric connection. Eventually, we establish conformal equations of Gauss curvature and Codazzi-Mainardi. [ABSTRACT FROM AUTHOR]

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R 4 having planar geodesics with respect to the affine metric. [ABSTRACT FROM AUTHOR]

In this paper we define a diagonal lift Dg of Riemannian metric g of manifold Mn to the tangent bundle of order two denoted by T²Mn of Mn, we associate to Dg its Levi-civita connection of T² M and we investigate applications of the diagonal lifts in the killing vectors and geodesics. [ABSTRACT FROM AUTHOR]

We deduce two necessary and sufficient conditions for a diffeomorphism $f\backslash :M\backslash to\; \backslash overline\; M$ of a Riemannian manifold $(M,g)$ onto a Riemannian manifold $(\backslash overline\; M,\backslash bar\; g)$ to be harmonic. Using the representation theory of groups, we define in an intrinsic way seven classes of such harmonic diffeomorphisms and partly describe the geometry of each class. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define a diagonal lift Dg of a Riemannian metric g of a manifold Mn to the tensor bundle T1q(Mn) of Mn, to associate with Dg an Levi–Civita connection of T1q(Mn) in a natural way and to investigate applications of the diagonal lifts. [Copyright &y& Elsevier]

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). [ABSTRACT FROM AUTHOR]