The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate …
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The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla…
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
The paper studies special solitons on Riemannian manifolds with specific vector fields.
problem Characterizing conformal and -Yamabe solitons with torse forming potential vector fields.
method Analyzing solitons under different connections (Riemannian, semi-symmetric, projective semi-symmetric) and developing examples.
result Characterizations and properties of conformal and -Yamabe solitons with torse forming vector fields.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isola…
Introduces canonical connections for sub-Riemannian manifolds with constant symbol.
problem Equivalence problem in sub-Riemannian geometry.
method Introduces canonical grading and compatible affine connection.
result Completely computed structures for contact manifolds of constant symbol.
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.
Defines and proves the uniqueness of a second natural connection on Riemannian Π-manifolds.
problem Characterizing and proving uniqueness of a natural connection on Riemannian Π-manifolds.
method Definition and proof of the second natural connection, proving its uniqueness and necessary/sufficient condition for coincidence with the first natural connection.
result Proves the uniqueness of the second natural connection on Riemannian Π-manifolds.
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Classifies Riemannian manifolds with specific torsion properties.
problem Classifying Riemannian manifolds with parallel, non-twistorial torsion.
method Classifies complete simply connected Riemannian manifolds with a metric connection having parallel torsion, non-zero vectorial component, and zero twistorial component.
result Classifies complete simply connected Riemannian manifolds with the specified torsion properties.
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.
problem Geometric proof of the Frobenius theorem on Riemannian manifolds.
method Introduces a new canonical connection and applies it to prove the Frobenius theorem.
result Geometric proof of the Frobenius theorem.
We build an analogue for the Levi-Civita connection on Riemannian manifolds for sub-Riemannian manfiolds modeled on the Heisenberg group. We demonstrate some geometric properties of this connection to justify our choice and show that this connection is unique in having these properties.
The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable li…
The paper characterizes metallic pseudo-Riemannian manifolds using conjugate connections and tensor structures.
problem Characterizing metallic pseudo-Riemannian manifolds.
method Using conjugate connections and tensor structures, the paper derives new characterizations and conditions for these manifolds.
result A necessary and sufficient condition for a non-integrable metallic pseudo-Riemannian manifold to be a quasi metallic pseudo-Riemannian manifold is derived.
The paper defines a new connection on sub-Riemannian manifolds and explores conditions for almost quasi-Sasakian manifolds to be Einstein.
problem Exploring new connections and conditions for specific types of manifolds.
method Defining a skew-symmetric connection and studying its properties on sub-Riemannian manifolds, and examining conditions for almost quasi-Sasakian manifolds to be Einstein.
result Sufficient conditions are found for an almost quasi-Sasakian manifold to be an Einstein manifold.
Some invariant tensors in two Naveira classes of Riemannian product manifolds are considered. These tensors are related with natural connections, i.e. linear connections preserving the Riemannian metric and the product structure.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…
Ambrose and Singer characterized connected, simply-connected and complete homogeneous Riemannian manifolds as Riemannian manifolds admitting a metric connection such that its curvature and torsion are parallel. The aim of this paper is to extend Ambrose-Singer Theorem to the general framework of locally homogeneous pse…
The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an…
The paper characterizes Riemannian manifolds using concircular vector fields and a connecting function.
problem Characterizing Riemannian manifolds using concircular vector fields.
method Introducing a connecting function that links concircular vector fields to potential functions.
result The connecting function is crucial for characterizing -sphere and Euclidean space.
New insights into metrizability of SO(3)-invariant connections, linking Riemann and Finsler structures.
problem Clarifying metrizability of SO(3)-invariant connections between Riemann and Finsler structures.
method Analyzing 4D SO(3)-invariant, Berwald-Finsler metrizable connections to find non-metric but metrizable connections.
result Identified classes of SO(3)-invariant connections that are not Levi-Civita connections for any pseudo-Riemannian metric but can still be metrized by Finsler functions.
The paper develops a non-transitive Cartan connection for sub-Riemannian manifolds.
problem Analyzing sub-Riemannian structures and their symmetries.
method Using Lie groupoids and non-transitive Cartan connections.
result A non-transitive Cartan connection is constructed for sub-Riemannian manifolds.
Defines Witt structures for pseudo-Riemannian manifolds.
problem No specific problem stated; focuses on new structure.
method Introduces a connection adapted to Witt structures.
result Revisits geodesics and symmetric spaces in new context.
Study extends Yang-Mills energy gap to Kähler surfaces.
problem Extend Yang-Mills energy gap to Kähler surfaces.
method Extend -energy gap to compact Kähler surfaces with generic Kähler metrics.
result All ASD connections on principal bundles over Kähler surfaces are irreducible.
Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifoldsmath.DG
On a Riemannian almost product manifold we consider a linear connection preserving the almost product structure and the Riemannian metric and having a totally skew-symmetric torsion. We determine the class of the manifolds admitting such a connection and prove that this connection is unique …
New method for curvature computation in sub-Riemannian geometry.
problem Computing curvature in sub-Riemannian manifolds.
method Using compatible affine connections and induced tensors.
result Universal Bonnet-Myers theorem for sub-Riemannian geometry.
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
The class W_1 of conformal Riemannian P-manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kaehler manifolds in almost Hermitian geometry. In the p…
Holonomy groups of K-contact sub-Riemannian manifolds are isomorphic.
problem Holonomy groups of K-contact sub-Riemannian manifolds.
method Proved isomorphism of holonomy groups under specific conditions.
result Holonomy groups of K-contact sub-Riemannian manifolds are isomorphic.
In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.
New connections defined for a specific geometric structure.
problem Characterizing manifolds with a paracontact structure.
method Introduced and studied a pair of associated Schouten-van Kampen affine connections.
result Curvature properties of the connections are obtained.
The study explores properties of metric connections with skew torsion and their curvature identities.
problem Investigating curvature properties of metric connections with skew torsion.
method Analyzing the curvature and torsion properties of metric connections with skew torsion.
result Necessary and sufficient conditions for a metric connection with skew torsion to satisfy the Riemannian first and second Bianchi identities are presented.
Extends index theorem to domain walls with discontinuous Riemannian connections.
problem Index theorem for domain walls with discontinuous Yang-Mills and Riemannian connections.
method Extension of index theorem to new conditions.
result Validates index theorem for more complex discontinuities.
We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-We…
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the …
In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing pr…
For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizi…
Study lift metrics and connections on tangent bundles of Riemannian manifolds.
problem Investigate geometric properties of tangent bundles and their lifts.
method Analyze lift metrics and connections on of , and study statistical and Codazzi couples.
result Prove a result on -Stein and Osserman structures on .
The study investigates properties of a specific Riemannian manifold with a semi-symmetric non-metric connection.
problem Characterizing properties of a Riemannian manifold with a semi-symmetric non-metric connection.
method Construction of a non-trivial example, proving manifold properties based on the metric being a gradient soliton or Yamabe soliton.
result A manifold with a semi-symmetric non-metric connection and gradient Ricci/Yamabe soliton is of constant curvature.
We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
The Prytz connectionsmath.DG
Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.
problem Historical use of Prytz planimeter to approximate areas.
method Sub-Riemannian geometry and connections/horizontal lifts.
result Mathematical description and analysis of Prytz planimeter.