### An explicit classification of 3-dimensional Riemannian spaces satisfying $R(X,Y)\xb7R=0$

Oldřich Kowalski (1996)

Czechoslovak Mathematical Journal

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Oldřich Kowalski (1996)

Czechoslovak Mathematical Journal

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David E. Blair, Alexander P. Stone (1971)

Annales de l'institut Fourier

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Let $M$ be an $(n+1)$-dimensional Riemannian manifold admitting a covariant constant endomorphism $h$ of the localized module of 1-forms with distinct non-zero eigenvalues. After it is shown that $M$ is locally flat, a manifold $N$ immersed in $M$ is studied. The manifold $N$ has an induced structure with $n$ of the same eigenvalues if and only if the normal to $N$ is a fixed direction of $h$. Finally conditions under which $N$ is invariant under $h$, $N$ is totally geodesic and the induced structure has vanishing...

Paolo Piccinni (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si considera la seconda forma fondamentale $\alpha $ di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano ${\nabla}^{2}\alpha $ - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

Mohamed Tahar Kadaoui Abbassi, Maâti Sarih (2005)

Archivum Mathematicum

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There is a class of metrics on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric $g$ [Kow-Sek1]. We call them “$g$-natural metrics" on $TM$. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on $TM$ from some quadratic forms on $OM\times {\mathbb{R}}^{m}$ to find metrics (not necessary...

Alexandru Kristály, Vicenţiu Rădulescu (2009)

Studia Mathematica

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Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem $-{\Delta}_{g}\omega +\alpha \left(\sigma \right)\omega =K\u0303(\lambda ,\sigma )f\left(\omega \right)$, σ ∈ M, ω ∈ H₁²(M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, ${\Delta}_{g}$ stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These...

Hiraku Nozawa, José Ignacio Royo Prieto (2014)

Annales de l’institut Fourier

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We show that any transversally complete Riemannian foliation $\mathcal{F}$ of dimension one on any possibly non-compact manifold $M$ is tense; namely, $M$ admits a Riemannian metric such that the mean curvature form of $\mathcal{F}$ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize...

Yaning Wang, Ximin Liu (2014)

Annales Polonici Mathematici

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We consider an almost Kenmotsu manifold ${M}^{2n+1}$ with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that ${M}^{2n+1}$ is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that ${M}^{2n+1}$ is ξ-Riemannian-semisymmetric. Moreover, if ${M}^{2n+1}$ is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove...

Miomir Stanković, Svetislav Minčić (2000)

Publications de l'Institut Mathématique

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Ugo Boscain, Mario Sigalotti (2006-2007)

Séminaire de théorie spectrale et géométrie

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Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally...