This paper, considers the fuzzy topological subsets, fuzzy topological spaces and introduces a novel concept of fuzzy Hausdorff space and fuzzy manifold space in this regards. Based on these concepts, we present a concept of fuzzy metric Hausdorff spaces and fuzzy metric manifold spaces via the notations of KM-fuzzy metric spaces. This study, generalises the concept of fuzzy metric space to union and product of fuzzy metric spaces in classical logic and in this regard investigates the some product of fuzzy metric fuzzy manifold spaces. We apply the notation of valued-level subsets and make a relation between of topological space, Husdorff space, manifold space and fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space. In final, we extended the fuzzy topological space, fuzzy Husdorff space and fuzzy manifold space to fuzzy metric topological space, fuzzy metric Husdorff space and fuzzy metric manifold space Indeed, this study analyses the notation of fuzzy metric manifold based on valued-level subset.

In this paper, we investigate pseudo-Riemannian manifolds those eigenvalues of the Weyl conformal Jacobi operators are constant on the unit sphere bundles. Using a result of [4], we give an explicit construction of conformally Osserman manifold which is not locally conformally flat.

Dimensionality reduction is an important task in machine learning, to simplify data mining, image processing, classification and visualization of high-dimensional data by mitigating undesired properties of high-dimensional spaces. manifold learning is a relatively new approach to nonlinear dimensionality reduction. Algorithms for manifold learning are based on the intuition that the dimensionality of many data sets may be artificially high and each data point can be described as a function of only a few underlying parameters. Using this tool, intrinsic parameters of the system database, which are main distinction factors of data sets, are recognized and all of them lie on a manifold that shows the real relationship of parameters. One of the successful applications of these methods is in image analysis field. By this approach, each image is a data in high dimensional space that the pixels are its dimensions. Because echocardiography images obtained from a patient are different in quantitative parameters such as heartbeat periodic motion and noise, image sets are reduced to two-dimensional space by a proper manifold learning. In this article, after mapping echocardiography images in two-dimensional space, by using LLE and Isomap algorithms, similar images placed side by side and the relationships between the images according to the cyclic property of heartbeat became evident. The Results showed the weakness of Isomap algorithm and power of LLE algorithm in preserving the relation between consecutive frames. De-noising is an important application which extracted from this research.

We classify the paracontact Riemannian manifolds that their Riemannian curvature satis es in the certain condition and we show that this classi cation holds for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X;  )
S = 0, where S is the Ricci tensor.

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