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Ramazan Sari - One of the best experts on this subject based on the ideXlab platform.
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on semi slant varvec xi perp riemannian submersions
Mediterranean Journal of Mathematics, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
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On Semi-Slant $${\varvec{\xi ^\perp }}$$-Riemannian Submersions
Mediterranean Journal of Mathematics, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
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On semi-slant $\xi^\perp-$Riemannian submersions
arXiv: Differential Geometry, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper to define and study semi-slant $\xi^\perp-$Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant $\xi^\perp-$Riemannian submersions, semi-invariant $\xi^\perp-$Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
Mehmet Akif Akyol - One of the best experts on this subject based on the ideXlab platform.
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on semi slant varvec xi perp riemannian submersions
Mediterranean Journal of Mathematics, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
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On Semi-Slant $${\varvec{\xi ^\perp }}$$-Riemannian Submersions
Mediterranean Journal of Mathematics, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper is to define and study semi-slant \(\xi ^\perp \)-Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant \(\xi ^\perp \)-Riemannian submersions, semi-invariant \(\xi ^\perp \)-Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
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On semi-slant $\xi^\perp-$Riemannian submersions
arXiv: Differential Geometry, 2017Co-Authors: Mehmet Akif Akyol, Ramazan SariAbstract:The aim of the present paper to define and study semi-slant $\xi^\perp-$Riemannian submersions from Sasakian Manifolds onto Riemannian Manifolds as a generalization of anti-invariant $\xi^\perp-$Riemannian submersions, semi-invariant $\xi^\perp-$Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base Manifold to be a locally Product Manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.
Mehmet Atçeken - One of the best experts on this subject based on the ideXlab platform.
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SLANT SUBManifoldS OF A RIEMANNIAN Product Manifold
Acta Mathematica Scientia, 2010Co-Authors: Mehmet AtçekenAbstract:Abstract In this article, the geometry of the slant subManifolds of a Riemannian Product Manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant subManifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant subManifolds in a Riemannian Product Manifold.
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SubManifolds of Riemannian Product Manifolds
Turkish Journal of Mathematics, 2005Co-Authors: Mehmet AtçekenAbstract:In this paper, we study the geometry of the semi-invariant subManifolds of a Riemannian Product Manifold. Fundamental properties of these type subManifolds such as the integrability of the distributions D, D\bot and mixed-geodesic property are studied. Finally, necessary and sufficient conditions are given on a semi-invariant subManifold of Riemannian Product Manifold to be D-geodesic and D\bot-geodesic.
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F-Invariant SubManifolds of Kaehlerian Product Manifold
Turkish Journal of Mathematics, 2004Co-Authors: Mehmet AtçekenAbstract:In this paper, the geometry of F-invariant subManifolds of a Kaehlerian Product Manifold is studied. The fundamental properties of these subManifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic subManifold and locally decomposable Riemannian Product Manifold.
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On Invariant SubManifolds of Riemannian Warped Product Manifold
Turkish Journal of Mathematics, 2003Co-Authors: Mehmet Atçeken, Bayram Şahin, Erol KiliçAbstract:In this paper, we generalize the geometry of the invariant subManifolds of Riemannian Product Manifold to the geometry of the invariant subManifolds of Riemannian warped Product Manifold. We investigate some properties of an invariant subManifolds of a Riemannian warped Product Manifold. We show that every invariant subManifold of the Riemannian warped Product Manifold is a Riemannian warped Product Manifold. Also, we give a theorem on the pseudo-umbilical invariant subManifold. Further, we obtain that integral Manifolds on an invariant subManifold are curvature-invariant subManifolds. Finally, we give a necessary condititon on a totally umbilical invariant subManifold to be totally geodesic.
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Semi-invariant subManifolds of Riemannian Product Manifold.
2003Co-Authors: Bayram Şahin, Mehmet AtçekenAbstract:In this paper, the geometry of subManifolds of a Riemannian Product Manifold is studied. Fundamental properties of these subManifolds are investigated such as integrability of distributions, totally umbilical semi-invariant subManifold. Finally, necessary and sucient conditions are given on a semi-invariant subManifold of a Riemannian Product Manifold to be a locally Riemannian Manifold.
Yui-man Lui - One of the best experts on this subject based on the ideXlab platform.
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using a Product Manifold distance for unsupervised action recognition
Image and Vision Computing, 2012Co-Authors: Stephen Ohara, Yui-man Lui, Bruce A. DraperAbstract:This paper presents a method for unsupervised learning and recognition of human actions in video. Lacking any supervision, there is nothing except the inherent biases of a given representation to guide grouping of video clips along semantically meaningful partitions. Thus, in the first part of this paper, we compare two contemporary methods, Bag of Features (BOF) and Product Manifolds (PM), for clustering video clips of human facial expressions, hand gestures, and full-body actions, with the goal of better understanding how well these very different approaches to behavior recognition produce semantically relevant clustering of data. We show that PM yields superior results when measuring the alignment between the generated clusters and the nominal class labeling of the data set. We found that while gross motions were easily clustered by both methods, the lack of preservation of structural information inherent to the BOF representation leads to limitations that are not easily overcome without supervised training. This was evidenced by the poor separation of shape labels in the hand gestures data by BOF, and the overall poor performance on full-body actions. In the second part of this paper, we present an unsupervised mechanism for learning micro-actions in continuous video streams using the PM representation. Unlike other works, our method requires no prior knowledge of an expected number of labels/classes, requires no silhouette extraction, is tolerant to minor tracking errors and jitter, and can operate at near real-time speed. We show how to construct a set of training ''tracklets,'' how to cluster them using the Product Manifold distance measure, and how to perform detection using exemplars learned from the clusters. Further, we show that the system is amenable to incremental learning as anomalous activities are detected in the video stream. We demonstrate performance using the publicly-available ETHZ Livingroom data set.
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Human Gesture Recognition on Product Manifolds
JOURNAL OF MACHINE LEARNING RESEARCH;13: 3297-3321 NOV 2012, 2012Co-Authors: Yui-man LuiAbstract:Action videos are multidimensional data and can be naturally represented as data tensors. While tensor computing is widely used in computer vision, the geometry of tensor space is often ignored. The aim of this paper is to demonstrate the importance of the intrinsic geometry of tensor space which yields a very discriminating structure for action recognition. We characterize data tensors as points on a Product Manifold and model it statistically using least squares regression. To this aim, we factorize a data tensor relating to each order of the tensor using Higher Order Singular Value Decomposition (HOSVD) and then impose each factorized element on a Grassmann Manifold. Furthermore, we account for underlying geometry on Manifolds and formulate least squares regression as a composite function. This gives a natural extension from Euclidean space to Manifolds. Consequently, classification is performed using geodesic distance on a Product Manifold where each factor Manifold is Grassmannian. Our method exploits appearance and motion without explicitly modeling the shapes and dynamics. We assess the proposed method using three gesture databases, namely the Cambridge hand-gesture, the UMD Keck body-gesture, and the CHALEARN gesture challenge data sets. Experimental results reveal that not only does the proposed method perform well on the standard benchmark data sets, but also it generalizes well on the one-shot-learning gesture challenge. Furthermore, it is based on a simple statistical model and the intrinsic geometry of tensor space.
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CVPR - Action classification on Product Manifolds
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Yui-man Lui, J. Ross Beveridge, Michael KirbyAbstract: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.
Michael Kirby - One of the best experts on this subject based on the ideXlab platform.
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CVPR - Action classification on Product Manifolds
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Yui-man Lui, J. Ross Beveridge, Michael KirbyAbstract: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.
