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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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TL;DR: In this paper, the authors investigated the cutoff dependence of the tensor/scalar ratio during inflation and found that the cutoff introduces an ambiguity in the choice of action for tensor and scalar perturbations, which in turn can affect this ratio.
60 citations
TL;DR: This work compares two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube and gets new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discretehypercube.
Abstract: We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.
60 citations
TL;DR: In this paper, the authors constructed all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries.
Abstract: We construct all independent local scalar monomials in the Riemann tensor at an arbitrary dimension, for the special regime of static spherically symmetric geometries. Compared to general spaces, their number is significantly reduced: the extreme example is the collapse of all invariants ~Weyl^k, to a single term at each k. The latter is equivalent to the Lovelock invariant L_k . Depopulation is less extreme for invariants involving rising numbers of Ricci tensors, and also depends on the dimension. The corresponding local gravitational actions and their solution spaces are discussed.
60 citations
TL;DR: The tensor decomposition addressed in this paper may be seen as a generalization of Singular Value Decomposition of matrices and how the decomposition can be recovered from eigenvector computation.
59 citations
TL;DR: In this article, a compactness theorem for the space of closed embedded f-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Emery Ricci curvature was proved.
Abstract: In this paper, we first prove a compactness theorem for the space of closed embedded f-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry–Emery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the f-Laplacian on a compact manifold with positive m-Bakry–Emery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the n-sphere, or the n-dimensional hemisphere. Finally, for a compact manifold with positive m-Bakry–Emery Ricci curvature and f-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if and only if the manifold is isometric to a Euclidean ball.
59 citations