Topic
Riemann curvature tensor
About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.
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TL;DR: In this paper, the authors show that the Laplacian flow will blow up at a finite-time singularity, so the flow will exist as long as the velocity of the flow remains bounded.
Abstract: We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on
$$\Lambda(x,t)=\left(|
abla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$$
will imply bounds on all covariant derivatives of Rm and T. (2). We show that $${\Lambda(x,t)}$$
will blow up at a finite-time singularity, so the flow will exist as long as $${\Lambda(x,t)}$$
remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.
86 citations
TL;DR: In this paper, the authors considered universal lower bounds on the volume of a Riemannian manifold, given in terms of the volumes of lower dimensional objects (primarily the lengths of geodesics).
Abstract: In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By ‘universal’ we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary, allows the survey to be reasonably short. Although, even in this limited case the authors have left out many interesting results.
85 citations
85 citations
TL;DR: In this article, a view of the relation between the continuum theory of defects in crystals and the mathematical theory of non-metric, non-Riemannian geometry is presented in the linear approximation.
85 citations
TL;DR: In this paper, it was shown that a Riemannian manifold (M, g) close enough to the round sphere in the C4 topology to have uniformly convex injectivity domains has a nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
Abstract: We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
85 citations