Theory of spacetime impetus

This article introduces the theory of spacetime impetus (SI). The theory unites Newtonian theory (NT) and the theory of general relativity (GR). To develop SI, we reformulated NT in spacetime and replaced the particle primitive in NT with the fragment of energy primitive in field theory. SI replaces Newton’s second law F = ma governing the motion of particles, where F, m, and a are, respectively, interaction force, mass, and acceleration, with the change equation P = k governing the motion of fragments of energy, where P and k are, respectively, action force and the curvature of a path in spacetime. To verify SI, we conducted three tests: Test 1 predicted the precession angles of Mercury and Jupiter, test 2 predicted the bending angle of light as it grazes the surface of the sun, and test 3 predicted the radius of the photon sphere. All three tests were in agreement with GR, the third corresponding to strong Riemannian curvature in GR. The equations of motion in SI are in terms of Cartesian coordinates and time and are relatively simple to solve. Undergraduate students in science and engineering and others with similar mathematical skills can validate the results for themselves.

Gradient plasticity in isotropic solids

We discuss a framework for the description of gradient plasticity in isotropic solids based on the Riemannian curvature derived from a metric induced by plastic deformation. This culminates in a flow rule in the form of a fourth-order partial differential equation for the plastic strain rate, in contrast to the second-order flow rules that have been proposed in alternative treatments of gradient plasticity in isotropic solids.

Curvature

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.

SOME NEW IDENTITIES FOR THE SECOND COVARIANT DERIVATIVE OF THE CURVATURE TENSOR

In this paper we study the second covariant derivative of Riemannian curvature tensor. Some new identities for the second covariant derivative are given. Namely, identities obtained by cyclic sum with respect to three indices are given. In the first case, two curvature tensor indices and one covariant derivative index participate in the cyclic sum, while in the second case one curvature tensor index and two covariant derivative indices participate in the cyclic sum.

Bi-slant submanifolds of an S manifold

PurposeBi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.Design/methodology/approachSome properties of Di-geodesic and Di-umbilical bi-slant submanifolds are examined.FindingsThe Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.Originality/valueThe topic is original, and the manuscript has not been submitted to any other journal.

Certain results on invariant submanifolds of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$-space

In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions $$\kappa ,\mu $$ κ , μ and $$\nu $$ ν behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -space to be totally geodesic under some conditions.

Rigidity of Complete Gradient Shrinkers with Pointwise Pinching Riemannian Curvature

Let M n , g , f be a complete gradient shrinking Ricci soliton of dimension n ≥ 3 . In this paper, we study the rigidity of M n , g , f with pointwise pinching curvature and obtain some rigidity results. In particular, we prove that every n -dimensional gradient shrinking Ricci soliton M n , g , f is isometric to ℝ n or a finite quotient of S n under some pointwise pinching curvature condition. The arguments mainly rely on algebraic curvature estimates and several analysis tools on M n , g , f , such as the property of f -parabolic and a Liouville type theorem.

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry

Most high-dimensional datasets are thought to be inherently low-dimensional—that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace–Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold’s embedding to its curvature using the Second Fundamental Form and the Gauss–Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

On the geometry of the tangent bundle with gradient Sasaki metric

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.