We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts (the Ricci tensor and the term proportional to the curvature scalar) with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, wit...

We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single braneworld solution [hep-th/9906064], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the ‘electric’ part of the 5-dimensional Weyl tensor, denoted by Eμν , gives the leading order correction to the conventional Einstein equations ...

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with deformed Sasaki metric. In this paper, firstly we investigate all forms of curvature tensors (Riemannian tensor, Ricci curvature, sectional scalar curvature). Secondly, study the geometry unit metric, where presented formulas Levi-Civita connection also

Provided certain basic rank requirements are met, we establish a converse of the classical fact that if A is symmetric, then RA is an algebraic curvature tensor. This allows us to establish a simultaneous diagonalization result in the event that three algebraic curvature tensors are linearly dependent. We use these results to establish necessary and sufficient conditions that a set of two or th...

Local curvature represents an important shape parameter of space curves which are well described by differential geometry. We have developed an estimator for local curvature of space curves embedded in n-dimensional (n-D) grey-value images. There is neither a segmentation of the curve needed nor a parametric model assumed. Our estimator works on the orientation field of the space curve. This or...

Einstein-Hilbert (EH) action can be separated into a bulk and a surface term, with a specific (‘‘holographic’’) relationship between the two, so that either can be used to extract information about the other. The surface term can also be interpreted as the entropy of the horizon in a wide class of spacetimes. Since EH action is likely to just the first term in the derivative expansion of an eff...

Suppose that (Mn, g) is an asymptotically flat Riemannian spin manifold of positive scalar curvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifold is always positive, and is zero if and only if the manifold is flat. This result suggests that there should be an inequality which bounds the Riemann tensor in terms of the total mass and implies that curvature must ...

This paper examines the action of GLN (R) on decomposable algebraic curvature tensors. The main result is that the structure group of a decomposable algebraic curvature tensor can only permute the subspaces into which the tensor decomposes: if the tensor decomposes into tensors Ri on Vi where V = ⊕k i=1 Vi, then for any matrix A in the structure group there exists a permutation σ such that A : ...

1. Manifolds: 1/7/14 1 2. Tangent and Cotangent Spaces: 1/9/14 3 3. Vector Fields, One-Forms, and Riemannian Metrics: 1/14/14 6 4. The Lie Bracket and Riemannian Connections: 1/16/14 8 5. Existence and Uniqueness of the Riemannian Connection: 1/21/14 10 6. Tensor Fields, Parallel Transport, and Holonomy: 1/23/14 13 7. The Riemann Curvature Tensor: 1/28/14 15 8. Flatness: 1/30/14 17 9. Symmetrie...

In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one-forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curv...