Левоинвариантная парасасакиева структура на групповой модели вещественного расширения плоскости де-Ситтера

We study symmetry properties of quaternionic K\"ahler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-K\"ahler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic K\"ahler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the $c$-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special K\"ahler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed $c$-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic K\"ahler metric on $SU(n,2)/S(U(n)\times U(2))$, which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet ($n=1$), for which we determine the full isometry group.

There is a series of papers devoted to the construction and investigation of local polynomial models of real submanifolds of complex space (see, e.g., [1]). Among the main properties of model surfaces, we can mention the following fact. The dimension of the local group of holomorphic symmetries of a germ does not exceed the dimension of the similar group for the tangent model surface of the germ. In the present note, this assertion is presented in a much stronger form. A faithful representation of the Lie algebra of infinitesimal automorphisms of a germ in the Lie algebra of the tangent model surface is constructed.

It is shown that the geometry of locally homogeneous multisymplectic manifolds(that is, smooth manifolds equipped with a closed nondegenerate form of degree$> 1$, which is locally homogeneous of degree $k$ with respect toa local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend thefamily of classical geometries possessing a similar property: symplectic,volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra ofinfinitesimal automorphisms, and on the study of the local properties ofHamiltonian vector fields on locally multisymplectic manifolds.In particular it is proved that thegroup of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.

This paper investigates the properties and behaviour of generalized βP-recurrent Finsler spaces with an emphasis on the covariant derivatives of Weyl's projective tensors, which include torsion, curvature, and deviation tensors. We define a GβP-recurrent space as a Finsler space where the second curvature tensor satisfies a specific recurrence condition involving non-zero covariant vector fields. Through a series of mathematical derivations, the paper explores the equivalence of different characterizations of the GβP-recurrent space, proving that the torsion tensor, its associate tensor, the P-Ricci tensor, and the curvature vector are non-vanishing. Further, we analyze the relationship between the covariant vector fields showing their dependence or independence on the direction argument. The study also includes the application of Berwald’s covariant derivative to the projective curvature tensor, concluding that these tensors exhibit generalized recurrence under certain conditions. The results are presented in a series of theorems that contribute to the deeper understanding of the geometric structure of - spaces.

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

In this paper, we consider a Lie group $G$ equipped with two left-invariant Riemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannian metrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangent Lie group $TG$. The Levi-Civita connection, tensor curvature, and sectional curvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, we give a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally, motivated by the recent work of D. N. Pham, using symplectic forms $\omega _1$ and $\omega _2$ on $G$ we define a symplectic form $\tilde{\omega}$ on $TG$.

Working within the recent formalism of Poisson–Riemannian geometry, we completely solve the case of generic spherically symmetric metric and spherically symmetric Poisson-bracket to find a unique answer for the quantum differential calculus, quantum metric and quantum Levi-Civita connection at semiclassical order . Here λ is the deformation parameter, plausibly the Planck scale. We find that are all forced to be central, i.e. undeformed at order λ, while for each value of r, t we are forced to have a fuzzy sphere of radius r with a unique differential calculus which is necessarily nonassociative at order . We give the spherically symmetric quantisation of the FLRW cosmology in detail and also recover a previous analysis for the Schwarzschild black hole, now showing that the quantum Ricci tensor for the latter vanishes at order λ. The quantum Laplace–Beltrami operator for spherically symmetric models turns out to be undeformed at order λ while more generally in Poisson–Riemannian geometry we show that it deforms to□f+λ2ωαβ(Ricγα−Sγ;α)(∇^βdf)γ+O(λ2)in terms of the classical Levi-Civita connection , the contorsion tensor S, the Poisson-bivector ω and the Ricci curvature of the Poisson-connection that controls the quantum differential structure. The Majid–Ruegg spacetime with its standard calculus and unique quantum metric provides an example with nontrivial correction to the Laplacian at order λ.

We prove that, on a generalized (multidimensional) Heisenberg group endowed with a left-invariant Sasakian structure, there exists a unique contact metric connection with skew torsion, invariant with respect to the group of automorphisms. The explicit expression of this connection in terms of the contact form and the metric tensor is obtained. It is shown that the torsion tensor and the curvature tensor are covariantly constant, and the sectional curvature varies between —1 and 0. It is proved that the obtained connection is a contact metric connection for any k-contact metric structure, therefore it is a contact metric connection for any Sasakian structure.

We propose a novel method of finding the classical limit of the matrix geometry. We define coherent states for a general matrix geometry described by a large-$N$ sequence of $D$ Hermitian matrices ${X}_{\ensuremath{\mu}}(\ensuremath{\mu}=1,2,\dots{},D)$ and construct a corresponding classical space as a set of all coherent states. When the classical space forms a smooth manifold, we also express various geometric objects on the classical space such as the metric, Levi-Civita connection, curvature and Poisson tensor, in terms of the matrix elements. This method provides a new class of observables in matrix models, which characterize geometric properties of matrix configurations.

In this thesis we introduce the S-theory. We apply the S-theory to variational problems of immersed Riemann surfaces, Elasticity theory and Spin theory. 
\nFrom classical mechanics it is known that geometric problems can be substantially simplified by symmetry considerations. The Noether theorem describes in this case how to find conservation laws from the invariance property of the observed energy. At first, the motivation was to give the space of immersions M = {f : M → R3}, for an oriented Riemannian manifold M of dimension less than or equal to three, an Euclidean invariant description. For geometric energies, i.e. energies which are invariant under Euclidean transformations, one would be able to derive conservation laws for the respective variational problems. It is a well known fact that immersed space curves are uniquely determined by their curvature functions (κ1,κ2,τ) up to Euclidean transformations. In the case of immersed surfaces and immersed 3-manifolds the situation becomes more complicated and leads to a non conformal deformation theory (S-Theorie). 
\nThe S-Theory builds the foundations of this thesis. Starting from a Riemannian manifold M and a reference metric ⟨ , ⟩ one can model any other Riemmanian metric g through a positive definite and self adjoint operator S via g = ⟨S,S⟩. 
\nThe operator S is an isometry, i.e. g = S∗⟨,⟩. From this point of view we compute the Levi- Civita connection with respect to g out of Levi-Civita connection of the reference metric and the operator S. 
\nFurther we introduce Spin bundles over two and three dimensional oriented Riemannian manifolds. Thereby Spin bundles, in contrast to most of the literature, have additionally a quaternionic structure. Many formulas become clearer and more accessible. We show that any Spin bundle Σ over (M, ⟨ , ⟩) has a unique Spin connection and compute via the S-Theory the deformed Spin connection with respect to the metric g. Eventually we consider the induced Spin bundle of an immersion f : M → R3 and compute the corresponding Spin connection and find a new interpretation of the Gauss-Codazzi equation. We introduce the Dirac operator and compute its deformation with respect to the metric g. 
\nWe are now able to give the space of immersed surfaces M = {f : M → R3} the desired geometric description. We compute the normal space of M and formulate the Noether theorem. As an application we compute conservation laws for the Willmore functional. 
\nFinally we introduce the intrinsic version of the famous distance-squared energy on a n-dimensional Riemannian manifold. We introduce the corresponding stress tensor and show that it’s closeness is a characterization for critical points of the energy. Thereby the S-theory provides a fundamental concept for the understanding of the stress tensor. In the case of a Riemann surface, the S-theory provides a canonical harmonic rotation 1-form for critical points of the distance-squared energy.

Levi-Civita connections from toral actions

By using a Riemannian metric on a differentiable manifold, the homogeneous lift metric is introduced on the (1, 1)-tensor bundle of the Riemannian manifold. Some geometric objects related to this metric, such as the Levi-Civita connection, Riemannian curvature tensor and sectional curvature are calculated. Also, a para-Nordenian structure on the (1, 1)-tensor bundle with this metric is constructed and interesting properties of this structure are studied.

We introduce the anisotropic tensor calculus, which is a way of handling tensors that depends on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.