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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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Proceedings ArticleDOI
13 Jun 2010
TL;DR: This paper argues that the underlying geometry of the tensor space is an important property for action classification and characterize a tensor as a point on a product manifold and perform classification on this space.
Abstract: Videos can be naturally represented as multidimensional arrays known as tensors. However, the geometry of the tensor space is often ignored. In this paper, we argue that the underlying geometry of the tensor space is an important property for action classification. We characterize a tensor as a point on a product manifold and perform classification on this space. First, we factorize a tensor relating to each order using a modified High Order Singular Value Decomposition (HOSVD). We recognize each factorized space as a Grassmann manifold. Consequently, a tensor is mapped to a point on a product manifold and the geodesic distance on a product manifold is computed for tensor classification. We assess the proposed method using two public video databases, namely Cambridge-Gesture gesture and KTH human action data sets. Experimental results reveal that the proposed method performs very well on these data sets. In addition, our method is generic in the sense that no prior training is needed.

137 citations

Journal ArticleDOI
TL;DR: This article presents an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor.
Abstract: Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, pen-and-ink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor. Our system converts each user specification into a basis tensor field and combines them with the input field to make an initial tensor field. However, such a field often contains unwanted degenerate points which cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite tensor indices. These operations provide control over the number and location of the degenerate points in the field. We observe that a tensor field can be locally converted into a vector field so that there is a one-to-one correspondence between the set of degenerate points in the tensor field and the set of singularities in the vector field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector fields. In addition, we adapt the image-based flow visualization technique to tensor fields, therefore allowing interactive display of tensor fields on surfaces. We demonstrate the capabilities of our tensor field design system with painterly rendering, pen-and-ink sketching of surfaces, and anisotropic remeshing

136 citations

Journal ArticleDOI
TL;DR: In this article, a covariant formalism for general multi-field systems is presented, which enables us to obtain higher order action of cosmological perturbations easily and systematically.
Abstract: We present a covariant formalism for general multi-field system which enables us to obtain higher order action of cosmological perturbations easily and systematically. The effects of the field space geometry, described by the Riemann curvature tensor of the field space, are naturally incorporated. We explicitly calculate up to the cubic order action which is necessary to estimate non-Gaussianity and present those geometric terms which have not yet been known before.

136 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202364
2022152
2021169
2020163
2019174
2018180