Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

SEMI‐RIEMANNIAN GEOMETRY With Applications to Relativity

A Riemannian manifold (M,g) is called a quasi Einstein manifold if for any coordinate system in M, its Ricci tensor S satisfes S_(ij) = ag_(ij) + bA_iA_j for some scalars a and b, where A(X)=g(X, p) for some unit vector p. This class of manifolds is a generalization of Einstein manifolds which are quasi Einstein manifolds whose b=0. In this paper, we will give examples of these manifolds and we will show that on each coordinate system, a and b are unique.

On Quasi-Einstein Manifolds Which Are Not Einstein


 Accurately measuring stellar parameters is a key goal to increase our understanding of the observable universe. However, current methods are limited by many factors, in particular, the biases and physical assumptions that are the basis for the underlying evolutionary or atmospheric models, those that these methods rely upon. Here we introduce our code spectrAl eneRgy dIstribution bAyesian moDel averagiNg fittEr (ARIADNE), which tackles this problem by using Bayesian Model Averaging to incorporate the information from all stellar models to arrive at accurate and precise values. This code uses spectral energy distribution fitting methods, combined with precise Gaia distances, to measure the temperature, log g, [Fe/H], AV, and radius of a star. When compared with interferometrically measured radii ARIADNE produces values in excellent agreement across a wide range of stellar parameters, with a mean fractional difference of only 0.001 ± 0.070. We currently incorporate six different models, and in some cases we find significant offsets between them, reaching differences of up to 550 K and 0.6 R⊙ in temperature and radius, respectively. For example, such offsets in stellar radius would give rise to a difference in planetary radius of 60%, negating homogeneity when combining results from different models. We also find a trend for stars smaller than 0.4-0.5 R⊙, which shows more work needs to be done to better model these stars, even though the overall extent is within the uncertainties of the interferometric measurements. We advocate for the use of ARIADNE to provide improved bulk parameters of nearby A to M dwarfs for future studies.

/pdf/ariadne-measuring-accurate-and-precise-stellar-parameters-qvzd6vt1.pdf

ARIADNE: Measuring accurate and precise stellar parameters through SED fitting

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2). Mathematics Subject Classification (2010). 53C15, 53C21, 53C25, 53C35.

Investigation on some classes of warped product manifolds

This paper characterizes the warping functions for a multiply generalized Robertson–Walker space-time to get an Einstein space M with a quarter-symmetric connection for different dimensions of M (i.e. (1). dim M = 2, (2). dim M ≥ 3) when all the fibers are Ricci flat. Then we have also computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of M (i.e. (1). dim M = 2, (2). dim M = 3, (3). dim M ≥ 4) and all the fibers are Ricci flat. In the last section, we have given two examples of multiply generalized Robertson–Walker space-time with respect to quarter-symmetric connection.

On Einstein Warped Products with a Quarter-Symmetric Connection

On Einstein Sequential Warped Product Spaces

In this paper, we have computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of M [i.e; (1). dimM = 2, (2). dimM = 3, (3). \({dim M \geq 4}\)] and all the fibers are Ricci flat.

On Ricci flat warped products with a quarter-symmetric connection

In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection associated to pair (Π, g). We introduce the notion of Einstein Poisson warped product space (M = B × f F, Π, g f ) (where Π = Π1 + Π2). Moreover, we show that if M is an Einstein Poisson warped product space with nonpositive scalar curvature and compact base B, J 1 is a field endomorphism on T*B satisfies , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where ) one can determine contravariant Einstein equations and the cosmological constant Λ corresponding to the contravariant Einstein equation G = −Λg. Moreover, it is shown that Einstein equation G = −Λg, induces the contravariant Einstein equation with cosmological constant Λ F on fiber space (F, g F , Π F ).

https://iopscience.iop.org/article/10.1088/1361-6382/abd7c0/pdf

Einstein Poisson warped product space

Abstract The volume of a tetrahedron is represented in terms of the twelve face angles, inradii of the faces of tetrahedron, circumradii of the faces and the radius of the sphere circumscribing the tetrahedron

Buddhadev Pal

Papers

On Einstein Warped Products with a Quarter-Symmetric Connection

On Einstein Sequential Warped Product Spaces

On Ricci flat warped products with a quarter-symmetric connection

Einstein Poisson warped product space

Volume of a tetrahedron revisited