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Journal articles on the topic 'Riemannian manifolds'

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Author: Grafiati

Published: 4 June 2021

Last updated: 31 July 2024

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1

Chaubey, Sudhakar, and Young Suh. "Riemannian concircular structure manifolds." Filomat 36, no. 19 (2022): 6699–711. http://dx.doi.org/10.2298/fil2219699c.

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In this manuscript, we give the definition of Riemannian concircular structure manifolds. Some basic properties and integrability condition of such manifolds are established. It is proved that a Riemannian concircular structure manifold is semisymmetric if and only if it is concircularly flat. We also prove that the Riemannian metric of a semisymmetric Riemannian concircular structure manifold is a generalized soliton. In this sequel, we show that a conformally flat Riemannian concircular structure manifold is a quasi-Einstein manifold and its scalar curvature satisfies the partial differential equation ?r = ?2r/?t2 + ?(n?1)?r/?t. To validate the existence of Riemannian concircular structure manifolds, we present some non-trivial examples. In this series, we show that a quasi-Einstein manifold with a divergence free concircular curvature tensor is a Riemannian concircular structure manifold.

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2

Sari, Ramazan, and Mehmet Akyol. "Hemi-slant ξ⊥-Riemannian submersions in contact geometry." Filomat 34, no. 11 (2020): 3747–58. http://dx.doi.org/10.2298/fil2011747s.

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M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

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3

Popov, Vladimir A. "Analytic Extension of Riemannian Analytic Manifolds of Small Dimension." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 2 (218) (June 23, 2023): 21–28. http://dx.doi.org/10.18522/1026-2237-2023-2-21-28.

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Abstract. An analytic extension of a locally given Riemannian analytic metric to a non-extendable Riemannian analytic manifold is considered. There are an infinite number of such extensions, and most of these extensions are very unnatural. The search for the most natural extensions leads to a generalization of the concept of completeness of a Riemannian manifold. It is possible to define a so called compressed manifold for metrics whose Lie algebra of Killing vector fields has no center. It is a universally attracting object in the category of all locally isometric Riemannian analytic manifolds. Morphisms of this category are locally isometric mappings 𝑓: 𝑀\𝑆, where 𝑆 is the set of fixed points of all local isometries of 𝑀 into itself that preserve orientation and Killing vector fields. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given. In contrast to a contracted manifold, a pseudocomplete manifold is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. M is non-extendable. There is no locally isometric covering mapping 𝑓: 𝑀 → 𝑁 where 𝑁 is a simply connected Riemannian analytic manifold, 𝑓(𝑀) is an open subset of 𝑁 not equal to 𝑁. In contrast to contracted manifolds, a pseudocomplete manifold is not unique in the class of locally isometric Riemannian analytic manifolds. Among the pseudocomplete manifolds, the most compressed regular pseudocomplete manifolds are defined. A classification of pseudocomplete manifolds of dimension 2 and 3 is given.

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4

Rovenski, Vladimir, Sergey Stepanov, and Irina Tsyganok. "The Mixed Scalar Curvature of a Twisted Product Riemannian Manifolds and Projective Submersions." Mathematics 7, no. 6 (June 10, 2019): 527. http://dx.doi.org/10.3390/math7060527.

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In the present paper, we study twisted and warped products of Riemannian manifolds. As an application, we consider projective submersions of Riemannian manifolds, since any Riemannian manifold admitting a projective submersion is necessarily a twisted product of some two Riemannian manifolds.

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5

Köprülü, Gizem, and Bayram Şahin. "Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers." International Journal of Geometric Methods in Modern Physics 18, no. 11 (June 29, 2021): 2150169. http://dx.doi.org/10.1142/s0219887821501693.

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The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

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6

ETAYO, FERNANDO, ARACELI DEFRANCISCO, and RAFAEL SANTAMARÍA. "Classification of pure metallic metric geometries." Carpathian Journal of Mathematics 38, no. 2 (February 28, 2022): 417–29. http://dx.doi.org/10.37193/cjm.2022.02.12.

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Metallic Riemannian manifolds with null trace and metallic Norden manifolds are generalizations of almost product Riemannian and almost golden Riemannian manifolds with null trace and almost Norden and almost Norden golden manifolds respectively. All these pure metrics geometries can be unified under the notion of α-metallic metric manifold. We classify this kind of manifolds in a consistent way with the well-known classifications of almost product Riemannian manifolds with null trace and almost Norden manifolds. We also characterize all classes of α-metallic metric manifolds by means of the first canonical connection which is a distinguished adapted connection.

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7

Ṣahin, Bayram. "Semi-invariant Submersions from Almost Hermitian Manifolds." Canadian Mathematical Bulletin 56, no. 1 (March 1, 2013): 173–83. http://dx.doi.org/10.4153/cmb-2011-144-8.

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AbstractWe introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semiinvariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

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8

PANTILIE, RADU. "Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 1 (July 2008): 141–51. http://dx.doi.org/10.1017/s0305004108001060.

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AbstractWe classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimensionn+2 to a Riemannian manifold of dimension 2, which can be factorised as ann-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

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9

Falbel, Elisha, Claudio Gorodski, and Michel Rumin. "Holonomy of Sub-Riemannian Manifolds." International Journal of Mathematics 08, no. 03 (May 1997): 317–44. http://dx.doi.org/10.1142/s0129167x97000159.

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A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).

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10

Gündüzalp, Yılmaz. "Anti-Invariant Semi-Riemannian Submersions from Almost Para-Hermitian Manifolds." Journal of Function Spaces and Applications 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/720623.

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We introduce anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds onto semi-Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a semi-Riemannian submersion, and check the harmonicity of such submersions. We also obtain curvature relations between the base manifold and the total manifold.

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11

Deng, Jialong. "Enlargeable length-structure and scalar curvatures." Annals of Global Analysis and Geometry 60, no. 2 (May 19, 2021): 217–30. http://dx.doi.org/10.1007/s10455-021-09772-7.

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AbstractWe define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

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12

Yang, Chun-Hao, and Baba C. Vemuri. "Nested Grassmannians for Dimensionality Reduction with Applications." Machine Learning for Biomedical Imaging 1, IPMI 2021 (March 1, 2022): 1–21. http://dx.doi.org/10.59275/j.melba.2022-234f.

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In the recent past, nested structure of Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested spheres. In this paper, we propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifolds. Common examples of homogeneous Riemannian manifolds include the spheres, the Stiefel manifolds, and the Grassmann manifolds. In particular, we focus on applying the proposed framework to the Grassmann manifolds, giving rise to the nested Grassmannians (NG). An important application in which Grassmann manifolds are encountered is planar shape analysis. Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grassmann manifold. Some salient features of our framework are: (i) it explicitly exploits the geometry of the homogeneous Riemannain manifolds and (ii) the nested lower-dimensional submanifolds need not be geodesic. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively. The proposed algorithms are compared with PGA via simulation studies and real data experiments and are shown to achieve a higher ratio of expressed variance compared to PGA.<br>The code is available at <a href='https://github.com/cvgmi/NestedGrassmann'>https://github.com/cvgmi/NestedGrassmann</a>

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13

Perrone, Domenico. "Curvature of K-contact Semi-Riemannian Manifolds." Canadian Mathematical Bulletin 57, no. 2 (June 14, 2014): 401–12. http://dx.doi.org/10.4153/cmb-2013-016-7.

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Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

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14

Ermolitski, A. A. "Riemannian regular $\sigma$-manifolds." Czechoslovak Mathematical Journal 44, no. 1 (1994): 57–66. http://dx.doi.org/10.21136/cmj.1994.128440.

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15

van Limbeek, Wouter. "Riemannian manifolds with local symmetry." Journal of Topology and Analysis 06, no. 02 (April 9, 2014): 211–36. http://dx.doi.org/10.1142/s179352531450006x.

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We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

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16

HASEBE, TAKAHIRO. "WHITE NOISE ANALYSIS ON MANIFOLDS AND THE ENERGY REPRESENTATION OF A GAUGE GROUP." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 04 (December 2010): 619–27. http://dx.doi.org/10.1142/s0219025710004243.

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The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility with the use of white noise analysis for compact Riemannian manifolds. In this paper we extend its technique to the noncompact Riemannian manifolds which have differential operators satisfying some conditions.

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17

Chiang, Yuan-Jen, and Hongan Sun. "Biharmonic maps on V-manifolds." International Journal of Mathematics and Mathematical Sciences 27, no. 8 (2001): 477–84. http://dx.doi.org/10.1155/s0161171201006731.

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We generalize biharmonic maps between Riemannian manifolds into the case of the domain being V-manifolds. We obtain the first and second variations of biharmonic maps on V-manifolds. Since a biharmonic map from a compact V-manifold into a Riemannian manifold of nonpositive curvature is harmonic, we construct a biharmonic non-harmonic map into a sphere. We also show that under certain condition the biharmonic property offimplies the harmonic property off. We finally discuss the composition of biharmonic maps on V-manifolds.

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18

Blažić, Novica, Neda Bokan, and Zoran Rakić. "Osserman pseudo-Riemannian manifolds of signature (2,2)." Journal of the Australian Mathematical Society 71, no. 3 (December 2001): 367–96. http://dx.doi.org/10.1017/s1446788700003001.

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AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

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19

Liu, Xi, Zhengming Ma, and Guo Niu. "Mixed Region Covariance Discriminative Learning for Image Classification on Riemannian Manifolds." Mathematical Problems in Engineering 2019 (February 28, 2019): 1–11. http://dx.doi.org/10.1155/2019/1261398.

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Covariance matrices, known as symmetric positive definite (SPD) matrices, are usually regarded as points lying on Riemannian manifolds. We describe a new covariance descriptor, which could improve the discriminative learning ability of region covariance descriptor by taking into account the mean of feature vectors. Due to the specific geometry of Riemannian manifolds, classical learning methods cannot be directly used on it. In this paper, we propose a subspace projection framework for the classification task on Riemannian manifolds and give the mathematical derivation for it. It is different from the common technique used for Riemannian manifolds, which is to explicitly project the points from a Riemannian manifold onto Euclidean space based upon a linear hypothesis. Under the proposed framework, we define a Gaussian Radial Basis Function- (RBF-) based kernel with a Log-Euclidean Riemannian Metric (LERM) to embed a Riemannian manifold into a high-dimensional Reproducing Kernel Hilbert Space (RKHS) and then project it onto a subspace of the RKHS. Finally, a variant of Linear Discriminative Analyze (LDA) is recast onto the subspace. Experiments demonstrate the considerable effectiveness of the mixed region covariance descriptor and the proposed method.

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20

Wang, Changliang. "Stability of Riemannian manifolds with Killing spinors." International Journal of Mathematics 28, no. 01 (January 2017): 1750005. http://dx.doi.org/10.1142/s0129167x17500057.

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Riemannian manifolds with nonzero Killing spinors are Einstein manifolds. Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1 ]. In this paper, we obtain a new proof for this stability result by using a Bochner-type formula in [X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161(1) (2005) 151–176; M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991) 815–844]. Moreover, existence of real Killing spinors is closely related to the Sasaki–Einstein structure. A regular Sasaki–Einstein manifold is essentially the total space of a certain principal [Formula: see text]-bundle over a Kähler–Einstein manifold. We prove that if the base space is a product of two Kähler–Einstein manifolds then the regular Sasaki–Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

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21

Popov, Vladimir A. "Locally Isometric Riemannian Analytic Spaces." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4-1 (216-1) (December 28, 2022): 55–64. http://dx.doi.org/10.18522/1026-2237-2022-4-1-55-64.

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Classes of locally isometric Riemannian analytic manifolds are studied. A generalization of the concept of completeness is given. We consider the Lie algebra 𝔤 of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra 𝔥 the simply connected Lie group 𝐺 corresponding to the Lie algebra 𝔤, and the subgroup 𝐻 corresponding to the Lie subalgebra 𝔥. In the absence of a center in the algebra 𝔤 the concept of a quasi-complete (compressed) manifold is introduced. An oriented Riemannian analytic manifold whose vector field algebra has zero center is said to be quasi-complete if it is non-extendable and does not admit non-trivial orientation-preserving and all Killing vector fields local isometries to itself. The main property of such a manifold is that it is unique in the class of all locally isometric Riemannian analytic manifolds, and any locally given isometry of this manifold 𝑀 into itself can be analytically extended to an isometry 𝑓: 𝑀 ≈ 𝑀. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given, which is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. 𝑀 is non-extendable. There is no locally isometric covering map f; M→N, where N is a simply connected Riemannian analytic manifold and f (M) is an open subset of N not equal to N.

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22

Sachdeva, Rashmi, Rupali Kaushal, Garima Gupta, and Rakesh Kumar. "Conformal slant submersions from nearly Kaehler manifolds." International Journal of Geometric Methods in Modern Physics 18, no. 06 (February 26, 2021): 2150088. http://dx.doi.org/10.1142/s0219887821500882.

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We study slant submersions and conformal slant submersions from nearly Kaehler manifolds onto Riemannian manifolds and investigate conditions for such maps to be totally geodesic maps. We also obtain conditions for a slant submersion and a conformal slant submersion from a nearly Kaehler manifold onto a Riemannian manifold to be a harmonic map and a harmonic morphism, respectively.

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23

Blaga, Adara-Monica, and Cristina-Elena Hretcanu. "Remarks on metallic warped product manifolds." Facta Universitatis, Series: Mathematics and Informatics 33, no. 2 (September 7, 2018): 269. http://dx.doi.org/10.22190/fumi1802269b.

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We characterize the metallic structure on the product of two metallic manifolds in terms of metallic maps and provide a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic. The particular case of product manifolds is discussed and an example of metallic warped product Riemannian manifold is provided.

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24

Rovenski, Vladimir. "Geometric Inequalities for a Submanifold Equipped with Distributions." Mathematics 10, no. 24 (December 14, 2022): 4741. http://dx.doi.org/10.3390/math10244741.

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The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.

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25

Bejancu, Aurel, and Reda Farran. "The scalar curvature of the tangent bundle of a Finsler manifold." Publications de l'Institut Math?matique (Belgrade) 89, no. 103 (2011): 57–68. http://dx.doi.org/10.2298/pim1103057b.

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Let Fm = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on the slit tangent bundle TM0 = TM \{0} of M. We express the scalar curvature ?~ of the Riemannian manifold (TM0,G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ?~ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of TM0. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ?~ satisfies the above condition.

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26

Bruce, Andrew James, and Janusz Grabowski. "Riemannian Structures on Z 2 n -Manifolds." Mathematics 8, no. 9 (September 1, 2020): 1469. http://dx.doi.org/10.3390/math8091469.

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Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

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27

Taştan, Hakan. "Lagrangian submersions from normal almost contact manifolds." Filomat 31, no. 12 (2017): 3885–95. http://dx.doi.org/10.2298/fil1712885t.

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We study Lagrangian submersions from Sasakian and Kenmotsu manifolds onto Riemannian manifolds. We prove that the horizontal distribution of a Lagrangian submersion from a Sasakian manifold onto a Riemannian manifold admitting vertical Reeb vector field is integrable, but the one admitting horizontal Reeb vector field is not. We also show that the horizontal distribution of a such submersion is integrable when the total manifold is Kenmotsu. Moreover, we give some applications of these results.

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28

Rustanov, Aligadzhi R., Elena A. Polkina, and Svetlana V. Kharitonova. "Identities of the Riemannian Curvature Tensor of Almost C(ʎ)-Manifolds." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4 (208) (December 23, 2020): 49–54. http://dx.doi.org/10.18522/1026-2237-2020-4-49-54.

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The geometry of the Riemannian curvature tensor of an almost C(λ)-manifold is studied. We have obtained several identities of the Riemannian curvature tensor of almost C(λ)-manifolds. Four additional identities are distinguished from these identities, on the basis of which four classes of almost C(λ)-manifolds are determined. A local classification of each of the distinguished classes of almost C(λ)-manifolds is obtained. It is proved that the set of almost C(λ)-manifolds of class R_1 coincides with the set of almost C(λ)-manifolds of class R_2, and it is also proved that the set of almost C(λ)-manifolds of class R_3 coincides with the set of almost C(λ)- manifolds of class R_4. We have found that an almost C(λ)-manifold, dimension greater than 3, is a manifold of class R_4 if and only if it is a cosymplectic manifold, i.e. when it is locally equivalent to the product of the Kähler manifold and the real line.

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29

Klepikova, S. V., and T. P. Makhaeva. "Mathematical Modeling in the Study of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with Isotropic Weyl Tensor." Izvestiya of Altai State University, no. 4(114) (September 9, 2020): 92–95. http://dx.doi.org/10.14258/izvasu(2020)4-14.

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It is known that a locally homogeneous manifold can be obtained from a locally conformally homogeneous (pseudo)Riemannian manifolds by a conformal deformation if the Weyl tensor (or the Schouten-Weyl tensor in the three-dimensional case) has a nonzero squared length. Thus, the problem arises of studying (pseudo)Riemannian locally homogeneous and locally conformally homogeneous manifolds, the Weyl tensor of which has zero squared length, and itself is not equal to zero (in this case, the Weyl tensor is called isotropic). One of the important aspects in the study of such manifolds is the study of the curvature operators on them, namely, the problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator. The problem of the prescribed values of the Ricci operator on 3-dimensional locally homogeneous Riemannian manifolds has been solved by O. Kowalski and S. Nikcevic. Analogous results for the one-dimensional and sectional curvature operators were obtained by D.N. Oskorbin, E.D. Rodionov, and O.P Khromova. This paper is devoted to the description of an example of studying the problem of the prescribed Ricci operator for four-dimensional locally homogeneous (pseudo) Riemannian manifolds with a nontrivial isotropy subgroup and isotropic Weyl tensor.

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Wei, Shihshu Walter, and Bing Ye Wu. "Generalized Hardy Type and Caffarelli–Kohn–Nirenberg Type Inequalities on Finsler Manifolds." International Journal of Mathematics 31, no. 13 (December 2020): 2050109. http://dx.doi.org/10.1142/s0129167x20501098.

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In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli–Kohn–Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.

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31

Peyghan, E., A. Tayebi, and L. Nourmohammadi Far. "On Twisted Products Finsler Manifolds." ISRN Geometry 2013 (July 10, 2013): 1–12. http://dx.doi.org/10.1155/2013/732432.

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On the product of two Finsler manifolds , we consider the twisted metric which is constructed by using Finsler metrics and on the manifolds and , respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study C-reducible and semi-C-reducible properties of this manifold. Then we obtain the Riemannian curvature and some of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature, mean Berwald curvature, and we find the relations between these objects and their corresponding objects on and . Finally, we study locally dually flat twisted product Finsler manifold.

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32

Choudhary, Ali, Mohd Siddiqi, Oğuzhan Bahadır, and Siraj Uddin. "Hypersurfaces of metallic Riemannian manifolds as k-Almost Newton-Ricci solitons." Filomat 37, no. 7 (2023): 2187–97. http://dx.doi.org/10.2298/fil2307187c.

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This research investigates k-Almost Newton-Ricci solitons (k-ANRS) embedded in a metallic Riemannian manifold Mn having the potential function ?. Furthermore, we prove geodesic and minimal conditions for hypersurfaces of metallic Riemannian manifolds. Beside this, we have explained some applications of metallic Riemannian manifold admitting k-Almost Newton-Ricci solitons.

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MANTICA, CARLO ALBERTO, and YOUNG JIN SUH. "PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS." International Journal of Geometric Methods in Modern Physics 10, no. 05 (April 3, 2013): 1350013. http://dx.doi.org/10.1142/s0219887813500138.

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In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds ( PS )n and pseudo-concircular symmetric manifolds [Formula: see text] is defined. This is named pseudo-Q-symmetric and denoted with ( PQS )n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat ( PQS )n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of ( PQS )n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a ( PQS )n scalar field space-time is considered, and interesting properties are pointed out.

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Akyol, Mehmet Akif, and Cem Sayar. "Pointwise hemi-slant Riemannian submersions." Ukrains’kyi Matematychnyi Zhurnal 75, no. 10 (October 24, 2023): 1299–316. http://dx.doi.org/10.3842/umzh.v75i10.7257.

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UDC 517.98 We introduce a new type of submersions, which is called <em>pointwise hemi-slant Riemannian submersions,</em> as a generalization of slant Riemannian submersions, hemi-slant submersions, and pointwise slant submersions from Kaehler manifolds onto Riemannian manifolds. We obtain some geometric interpretations of this kind of submersions with respect to the total manifold, the base manifold, and the fibers. Moreover, we present non-trivial illustrative examples in order to demonstrate the existence of submersions of this kind. Finally, we obtain some curvature equalities and inequalities with respect to a certain basis.

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Noyan, Esra, and Yılmaz Gündüzalp. "Anti-invariant and Clairaut anti-invariant pseudo-Riemannian submersions in para-Kenmotsu geometry." Filomat 37, no. 24 (2023): 8247–59. http://dx.doi.org/10.2298/fil2324247n.

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In this paper, we describe anti-invariant and Clairaut anti-invariant pseudo-Riemannian submersions (AIPR and CAIPR submersions, respectively, briefly) from para-Kenmotsu manifolds onto Riemannian manifolds. We introduce new Clairaut circumstances for anti-invariant submersions whose total space is para-Kenmotsu manifold. Also, we offer a obvious example of CAIPR submersion.

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Stoica, Ovidiu Cristinel. "The geometry of warped product singularities." International Journal of Geometric Methods in Modern Physics 14, no. 02 (January 18, 2017): 1750024. http://dx.doi.org/10.1142/s0219887817500244.

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In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

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Dokuzova, Iva. "Four-dimensional Riemannian product manifolds with circulant structures." Studia Universitatis Babes-Bolyai Matematica 68, no. 2 (June 13, 2023): 439–48. http://dx.doi.org/10.24193/subbmath.2023.2.17.

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"A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, is considered. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant, and it acts as an isometry with respect to the metric. The Riemannian product manifold associated with the considered manifold is studied. Conditions for the metric, which imply that the Riemannian product manifold belongs to each of the basic classes of Staikova-Gribachev's classi cation, are obtained. Examples of such manifolds are given."

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38

Osgood, Brad, and Dennis Stowe. "Riemannian manifolds." Duke Mathematical Journal 67, no. 1 (July 1992): 57–99. http://dx.doi.org/10.1215/s0012-7094-92-06704-4.

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Benci, Vieri, and Fabio Giannoni. "Riemannian manifolds." Duke Mathematical Journal 68, no. 2 (November 1992): 195–215. http://dx.doi.org/10.1215/s0012-7094-92-06808-6.

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40

Doi, Shin-ichi. "Riemannian manifolds." Duke Mathematical Journal 82, no. 3 (March 1996): 679–706. http://dx.doi.org/10.1215/s0012-7094-96-08228-9.

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Farran, H. R., E. El-Kholy, and S. A. Robertson. "Foldings of star manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, no. 6 (1993): 1011–16. http://dx.doi.org/10.1017/s0308210500029681.

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SynopsisThis paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.

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LI, XUE-MEI, and FENG-YU WANG. "ON THE COMPACTNESS OF MANIFOLDS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, supp01 (September 2003): 29–38. http://dx.doi.org/10.1142/s0219025703001249.

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It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

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Rovenski, Vladimir. "On isometric immersions of sub-Riemannian manifolds." Filomat 37, no. 25 (2023): 8543–51. http://dx.doi.org/10.2298/fil2325543r.

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We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.

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Rovenski, Vladimir. "Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds." Mathematics 11, no. 20 (October 21, 2023): 4377. http://dx.doi.org/10.3390/math11204377.

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Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold.

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Stepanov, S. E., I. I. Tsyganok, and V. Rovenski. "On conformal transformations of metrics of Riemannian paracomplex manifolds." Differential Geometry of Manifolds of Figures, no. 52 (2021): 117–22. http://dx.doi.org/10.5922/0321-4796-2020-52-11.

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A 2n-dimensional differentiable manifold M with -structure is a Riemannian almost paracomplex manifold. In the present paper, we consider conformal transformations of metrics of Riemannian paracomplex manifolds. In particular, a number of vanishing theorems for such transformations are proved using the Bochner technique.

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Chen, Bang-Yen, Sharief Deshmukh, and Amira A. Ishan. "On Jacobi-Type Vector Fields on Riemannian Manifolds." Mathematics 7, no. 12 (November 21, 2019): 1139. http://dx.doi.org/10.3390/math7121139.

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In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

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Manev, Mancho. "Pairs of Associated Yamabe Almost Solitons with Vertical Potential on Almost Contact Complex Riemannian Manifolds." Mathematics 11, no. 13 (June 26, 2023): 2870. http://dx.doi.org/10.3390/math11132870.

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Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed.

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Angers, Jean-Francois, and Peter T. Kim. "Symmetry and Bayesian Function Estimation1." Calcutta Statistical Association Bulletin 56, no. 1-4 (March 2005): 57–80. http://dx.doi.org/10.1177/0008068320050504.

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Summary This paper develops Bayesian function estimation on compact Riemannian manifolds. The approach is to combine Bayesian methods along with aspects of spectral geometry associated with the Laplace-Beltrami operator on Riemannian manifolds. Although frequentist nonparametric function estimation in Euclidean space abound, to date, no attempt has been made with respect to Bayesian function estimation on a general Riemannian manifold. The Bayesian approach to function estimation is very natural for manifolds because one can elicit very specific prior information on the possible symmetries in question . One can then establish Bayes estimators that possess built in symmetries. A detailed analysis for the 2–sphere is provided.

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Atçeken, Mehmet. "Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds." Mathematical Problems in Engineering 2009 (2009): 1–16. http://dx.doi.org/10.1155/2009/621625.

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We show that there exist no proper warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we consider warped product semi-invariant submanifolds in the form by reversing two factor manifolds and . We prove several fundamental properties of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a general inequality for an arbitrary warped product semi-invariant submanifold. After then, we investigate warped product semi-invariant submanifolds in a general almost paracontact Riemannian manifold which satisfy the equality case of the inequality.

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Agrachev, Andrei, Ugo Boscain, Robert Neel, and Luca Rizzi. "Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 3 (2018): 1075–105. http://dx.doi.org/10.1051/cocv/2017037.

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We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.

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