arXiv research

The study examines Riemann solitons and almost solitons on specific Kenmotsu manifolds.

We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…

Groupoids help define Riemann sums on manifolds.

The article investigates almost Riemann solitons and gradient almost Riemann solitons in a specific type of manifold.

Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold $M$ is equi…

Paper examines conditions making Riemann solitons trivial and estimates their scalar curvature.

Study of solitons in a specific type of contact metric manifold.

We develop Riemannian Stein Variational Gradient Descent (RSVGD), a Bayesian inference method that generalizes Stein Variational Gradient Descent (SVGD) to Riemann manifold. The benefits are two-folds: (i) for inference tasks in Euclidean spaces, RSVGD has the advantage over SVGD of utilizing information geometry, and …

Study on existence of balanced metrics on non-Kähler manifolds.

Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…

The study shows how certain surfaces can be filled by hyperbolic manifolds.

This paper proves positivity of Riemann-Roch polynomials for hyperkähler manifolds.

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of th…

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…

Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds.

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-rec…

Study on solitons in deformed Kenmotsu manifolds with specific vector fields.

We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

Exact diameter found for some Riemann surfaces.

Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection $\nabla$ (that is invariant under the group of automorphisms of V) is defined on every connected component of M, which in this way beco…

We prove that every suitable $4$-manifold with $b_1=0$ and with an embedded Riemann surface of genus $2$ is of simple type. We find a relationship between the basic classes of two of these $4$-manifolds and those of the connected sum along the Riemann surface.

Develops a new family of signature-changing models on metric manifolds.

Defines new invariants for Riemann-Finsler manifolds, generalizing Preissman's theorem.

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

Study of Tannakian categories for integrable connections on Kaehler manifolds.

In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.

The paper generalizes Riemann curvature for manifolds with discontinuous metrics.

Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings …

Study on almost Riemann solitons with gradient or torse-forming vector fields.

Local fractional derivatives affect Riemann curvature tensor to zero.

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli spa…

A new metric tensor improves Riemann manifold Monte Carlo for Bayesian models.

Study SO(3) vortices over orbifold Riemann surfaces and monopoles.

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Formula calculates Riemann-Roch number for singular symplectic quotients.

In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultima…

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origin…

The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

Develops equivariant Chern characters for coherent sheaves with group actions.

We generalized Xiang, Qi and Wei's results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the co…

The study identifies holomorphic sections on jet spaces of the Riemann sphere.

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two $4$-manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the $4$-manifold resulting as a connected sum along the surface. We…

In this paper we show that space of spatial polygons in semi riemann space gives a Kahler manifold. We describe the tangent space and almost complex structure which has many computational advantages.

Formula derived for Bott-Chern classes in complex blow-ups.

The paper extends Riemann-Hilbert correspondence to foliations.