On the Koszul formula in noncommutative geometry

In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M 1 M 2 and M be manifolds with dimensions n 1, n 2 and n, respectively. Also let (M l, g 1), (M 2, g 2 and (M, g) be semi-Riemannian manifolds of indices v l, v 2 and v with Levi-Civita connections \(\mathop \nabla \limits^1 ,\mathop \nabla \limits^2 \) and ∇, respectively, unless otherwise stated.

Let f:M→ℝ be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:M→G m (ℝ n ) into the Grassmannian G m (ℝ n ) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ℝ n . The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G m (ℝ n ) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.

<p style='text-indent:20px;'>In this paper we look at the question of integrability, or not, of the two natural almost complex structures <inline-formula><tex-math id="M1">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> defined on the twistor space <inline-formula><tex-math id="M2">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> of an even-dimensional manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> with additional structures <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula> a <inline-formula><tex-math id="M6">\begin{document}$ g $\end{document}</tex-math></inline-formula>-connection. We measure their non-integrability by the dimension of the span of the values of <inline-formula><tex-math id="M7">\begin{document}$ N^{J^\pm_\nabla} $\end{document}</tex-math></inline-formula>. We also look at the question of the compatibility of <inline-formula><tex-math id="M8">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> with a natural closed <inline-formula><tex-math id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-form <inline-formula><tex-math id="M10">\begin{document}$ \omega^{J(M, g, \nabla)} $\end{document}</tex-math></inline-formula> defined on <inline-formula><tex-math id="M11">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M12">\begin{document}$ (M, g) $\end{document}</tex-math></inline-formula> we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection <inline-formula><tex-math id="M13">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula>. In all cases <inline-formula><tex-math id="M14">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> is a bundle of complex structures on the tangent spaces of <inline-formula><tex-math id="M15">\begin{document}$ M $\end{document}</tex-math></inline-formula> compatible with <inline-formula><tex-math id="M16">\begin{document}$ g $\end{document}</tex-math></inline-formula>. In the case <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.</p>

In the present chapter we provide most of the prerequisites for reading the rest of the book. In the first two sections we present a review of vector bundles and introduce the main differential operators: Lie derivative, exterior differential, linear connection, general connection. Distributions on manifolds (known as non-holonomic spaces in classical terminology) are then introduced and studied by using both methods of vector fields and of differential 1-forms. We give here the characterization for the existence of a transversal distribution to a foliation, which is found to be very useful in Chapters 4 and 5 for a general study of lightlike submanifolds. In the last two sections we deal with semi-Riemannian manifolds and lightlike manifolds. While the geometry of a semi-Riemannian manifold is fully developed by using the Levi-Civita connection we stress the role of the radical distribution in studying the geometry of a lightlike manifold. The main formulas and results are expressed by using both the invariant form and the index form.

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Using Palatini’s formalism extended in a plausible fashion to the recent MS gp-Supergravity (TerKazarian, 2023c, 2024b), subject to certain rules, we reinterpret a flat MS gp-SG theory with Weitzenb¨ock torsion as the quantum field theory of Master-Space Teleparallel Supergravity (MS gp-TSG), having the gauge translation group in tangent bundle. Here the spin connection represents only inertial effects, but not gravitation at all. In order to recover the covariance, we introduce a 1-form of the Yang–Mills connection assuming values in the Lie algebra of the translation group. The Hilbert action vanishes and the gravitino action loses its spin connections, so that the accelerated reference frame has Weitzenb¨ock torsion induced by gravitinos. Due to the soldered character of the tangent bundle, torsion presents also the anholonomy of the translational covariant derivative. The gauge invariance of the tetrad provides torsion invariance under gauge transformations. The role of the Cartan-Killing metric usually comes, when it exists, from its being invariant under the group action. Here it does not exist, but we use the invariant Lorentz metric of Minkowski spacetime in its stead. The action of MS gp-TSG is invariant under local translations, under local super symmetry transformations and by construction is invariant under local Lorentz rotations and under diffeomorphisms. So that this action is invariant under the Poincar´e supergroup and under diffeomorphisms. We show the equivalence of the Teleparallel Gravity action with Hilbert action, which proves that the immediate cause of the fictitious Riemann curvature for the LeviCivita connection arises entirely due to the inertial properties of the Lorentz-rotated frame of interest. The curvature of Weitzenb¨ock connection vanishes identically, but for a tetrad involving a non-trivial translational gauge potential, the torsion is non-vanishing. We consider Weitzenb¨ock connection a kind of dual of the Levi-Civita connection, which is a connection with vanishing torsion, and non-vanishing fictitious curvature. The Weitzenb¨ock connection defines the acceleration through force equation, with torsion (or contortion) playing the role of force.

Diameter estimate for closed manifolds with positive scalar curvature

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 overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula for the standard Levi-Civita connection, a new point of view of the cosmological constant as a very small mass for the graviton of around $10^{-33}$ev, and a weakening of metric-compatibility in the presence of torsion. The same mechanism also provides a new construction for quantum bimodule connections on quantum spacetimes and a new approach to the quantum Ricci tensor.

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We

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.

We establish a one-to-one correspondence between a class of strictly almost Kähler metrics on the one hand and Lorentzian pp-wave spacetimes on the other; the latter metrics are well known in general relativity, where they model radiation propagating at the speed of light. Specifically, we construct families of complete almost Kähler metrics by deforming pp-waves via their propagation wave vector. The almost Kähler metrics we obtain exist in all dimensions \(2n \ge 4\), and are defined on both \(\mathbb {R}^{2n}\) and \({\mathbb {S}}^1\times {\mathbb {S}}^1 \times M\), where M is any closed almost Kähler manifold; they are not warped products, they include noncompact examples with constant negative scalar curvature, and all of them have the property that their fundamental 2-forms are also co-closed with respect to the Lorentzian pp-wave metric. Finally, we further deepen this relationship between almost Kähler and Lorentzian geometry by utilizing Penrose’s “plane wave limit,” by which every spacetime has, locally, a pp-wave metric as a limit: using Penrose’s construction, we show that in all dimensions \(2n \ge 4\), every Lorentzian metric admits, locally, an almost Kähler metric of this form as a limit.

Smooth manifolds with degenerate symmetric bilinear form of constant signature on tangent spaces are studied. Such manifolds are used in physical theories of unified interactions including gravitation. Necessary and sufficient conditions for the existence of a Levi-Civita connection on such a manifold are given. Bibliography: 6 titles.

Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

A Cartan manifold is a smooth manifold M whose slit cotangent bundle 0 * M T is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric ij g in the vertical bundle over 0 * M T and using it, a Sasaki type metric on 0 * M T is constructed. A natural almost complex structure is also defined by K on 0 * M T in such a way that pairing it with the Sasaki type metric an almost Kahler structure is obtained. In this paper we deform ij g to a pseudo-Riemannian metric ij G and we define a corresponding almost complex Kahler structure. We determine the Levi-Civita connection of G and compute all the components of its curvature. Then we prove that if the structure