N. G. Marchuk
We improve the concept of our previous paper "Dirac type tensor equations with nonabelian gauge symmetries on pseudo-Riemannian space" and present a new compact formula for the tensor $B_μ$.
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We suggest a so-called Dirac type tensor equation with nonabelian gauge symmetry on pseudo-Riemannian space. This equation reproduce some of the properties of spinor Dirac equation. A geometrical interpretation of results in terms of Riemannian geometry is given.
In this paper we extend the Cartan's approach of Riemannian normal coordinates and show that all n-dimensional pseudo-Riemannian metrics are conformal to a flat manifold, when, in normal coordinates, they are well-behaved in the origin and in its neighborhood. We show that for this condition all n-dimensioanl pseudo-Riemannian metrics can be embedded in a hyper-cone of an n+2-dimensional flat manifold. Based on the above conditions we show that each n-dimensional pseudo-Riemannian manifolds is conformal to an n-dimensional manifold of constant curvature. As a consequence of geometry, without p…
We generalize for pseudo-Riemannian metrics a classical result of Gallot and Tanno and use it to reprove a recent result of Alekseevsky, Cortes, Galaev and Leistner that decomposable cones over complete closed pseudo-Riemannian manifolds do not exist.
We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.
The traceless Ricci tensor $C_{ab}$ in 4-dimensional pseudo-Riemannian spaces equipped with the metric of the neutral signature is analyzed. Its algebraic classification is given. This classification uses the properties of $C_{ab}$ treated as a matrix. The Petrov-Penrose types of Plebański spinors associated with the traceless Ricci tensor are given. Finally, the classification is compared with a similar classification in the complex case.
The invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by non-trivial Killing tensors of valence two defined in the Euclidean and Minkowski planes. Illustrative examples are provided.
In this paper, we have reintroduced a new approach to conformal geometry developed and presented in two previous papers, in which we show that all n-dimensional pseudo-Riemannian metrics are conformal to a flat n-dimensional manifold as well as an n-dimensional manifold of constant curvature when Riemannian normal coordinates are well-behaved in the origin and in their neighborhood. This was based on an approach developed by French mathematician Elie Cartan. As a consequence of geometry, we have reintroduced the classical and quantum angular momenta of a particle and present new interpretation…
We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a bicovariant bimodule and its cocycle deformations are in one to one correspondence.
We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.
We study the higher order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r,s) for certain values of (r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher order Osserman manifolds.
We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.
We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.
We define the notion of Witt structure on the tangent bundle of a pseudo-Riemannian manifold and we introduce a connection adapted to a such structure. The notions of geodesics and symmetric spaces are revisited in this setting and applications are given in the special cases of Robinson and Fefferman manifolds.
In this note we establish the positivity of Green's functions for a class of elliptic differential operators on closed, Riemannian manifolds.
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
We discuss the role of the pseudo-Riemannian structure of the finite spectral triple for the family of Pati-Salam models. We argue that its existence is a very restrictive condition that separates leptons from quarks, and restricts the whole family of Pati-Salam models into the class of generalized Left-Right Symmetric Models.
I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing.
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical results of Thomsen.
We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classica…
Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard Model spectral triple in this formalism and interpret the physical symmetry preserving the lepton number as a shadow of a finite pseudo-Riemannian structure.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces
Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of $\mathcal{I}$-degenerate pseudo-Riemannian spaces, i.e., spaces having identical polynomial curvature invariants. In particular, we show that all the known examples of $\mathcal{I}$-degenerate pseudo-Riemannian spaces possess such vector fields.
We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly nonproportional (i.e., the minimal polynomial of $g^{iα} \bar g_{αj}$ coincides with the characteristic polynomial) at least at one point, then they have constant curvature.
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superin…
We prove the following statement: Let g be a light-line-complete pseudo-Riemannian Einstein metric of indefinite signature on a connected (n>2)-dimensional manifold M. Assume that a conformally equivalent metric is also Einstein. Then, the metrics are proportional with a constant coefficient. If in addition the manifold is closed, the assumption that the manifold is light-line-complete could be omitted.
We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions less than 7. Finally, we observe that every finitely generated torsion-free 2-step nilpotent group can be realized as the fundamental group of a complete flat pseudo-Riemannian manifold with abelian linear holonomy.
In this paper a thorough study of the normal form and the first integrability conditions arising from {\em bi-conformal vector fields} is presented. These new symmetry transformations were introduced in {\em Class. Quantum Grav.}\textbf{21}, 2153-2177 and some of their basic properties were addressed there. Bi-conformal vector fields are defined on a pseudo-Riemannian manifold through the differential conditions $\lie P_{ab}=ϕP_{ab}$ and $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$ and $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$. One of the main re…
We give necessary and sufficient conditions for Riemannian maps to be biharmonic. We also define pseudo umbilical Riemannian maps as a generalization of pseudo-umbilical submanifolds and show that such Riemannian maps put some restrictions on the base manifolds.
We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).
We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.
Pseudo-Riemannian and Hessian Geometry Related to Monge-Ampère Structures
该研究探究四维$T^*M$上Monge-Ampère结构对应的伪黎曼度量性质，给出由满足Plücker嵌入方程的六个系数参数化的Ricci平坦解族，还研究了二维$M$上伪度量拉回对应的Hessian结构。
The curvature tensor of a pseudo-Riemannian metric, and its covariant derivatives, satisfy certain identities that hold on any manifold of dimension less or equal than $n$. In this paper, we re-elaborate recent results by Gilkey-Park-Sekigawa regarding $p$-covariant dimensional curvature identities, for $p=0,2$. To this end, we use the classical theory of natural operations, that allows us to simplify some arguments and to generalize the description of Gilkey-Park-Sekigawa. Thus, our main result describes the first space of $p$-covariant dimensional curvature identities, for any even $p$.
We consider arbitrary-dimensional pseudo-Riemannian spaces of signature $(k,k+m)$. We introduce a boost-weight decomposition and define a number of algebraic properties (e.g., the ${\bf S}_i$- and ${\bf N}$-properties) and present a boost-weight decomposition to classify the Weyl tensors of arbitrary signature and discuss degenerate algebraic types (e.g., VSI spaces). We consider the four dimensional neutral signature space as an illustration.
The pseudo-Riemannian spectral zeta function density on scattering manifolds
该论文研究混合符号伪黎曼散射流形上拉普拉斯型算子复幂的谱zeta密度，证明其亚纯延拓并将极点与局部几何不变量关联，方法结合费曼预解估计与Hadamard参数化分析。
In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the ${\bf S}_i$- and ${\bf N}$-properties, and show that if the curvature tensors of the space possess the ${\bf N}$-property then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null-congruenc…
In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=ϕP_{ab}$, $\lieΠ_{ab}=χΠ_{ab}$ where $P_{ab}$, $Π_{ab}$ are orthogonal and complementary projectors with respect to the metric tensor $\rmg_{ab}$ and $\lie$ is the Lie derivative. In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of {\em conformally sep…
Existence results of Spin$(2,n-1)_0$-pseudo-Riemannian cobordisms
本文刻画了承载$(2,n-1)$型非退化伪黎曼度量、且边界具时间可定向 Lorentz 度量的 Spin$(n+1)$维配边的存在条件，并计算相应配边群。
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