scholarly journals Bi-f-harmonic curves and hypersurfaces

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5167-5180
Author(s):  
Selcen Perktaş ◽  
Adara Blaga ◽  
Feyza Erdoğan ◽  
Bilal Acet
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Harmonic Maps ◽  
Biharmonic Maps

In the present paper, we study bi-f-harmonic maps which generalize not only f-harmonic maps, but also biharmonic maps. We derive bi-f-harmonic equations for curves in the Euclidean space, unit sphere, hyperbolic space, and for hypersurfaces of Riemannian manifolds.

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

TRANSVERSALLY BIHARMONIC MAPS BETWEEN FOLIATED RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 19 (08) ◽  
pp. 981-996 ◽  
Author(s):  
YUAN-JEN CHIANG ◽  
ROBERT A. WOLAK
Keyword(s):  
Riemannian Manifolds ◽  
Positive Constant ◽  
Conservation Law ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Biharmonic Maps ◽  
Foliated Manifold ◽  
Transverse Curvature ◽  
Biharmonic Map ◽  
Foliated Manifolds

We generalize the notions of transversally harmonic maps between foliated Riemannian manifolds into transversally biharmonic maps. We show that a transversally biharmonic map into a foliated manifold of non-positive transverse curvature is transversally harmonic. Then we construct examples of transversally biharmonic non-harmonic maps into foliated manifolds of positive transverse curvature. We also prove that if f is a stable transversally biharmonic map into a foliated manifold of positive constant transverse sectional curvature and f satisfies the transverse conservation law, then f is a transversally harmonic map.

scholarly journals Biharmonic submanifolds in3-dimensional(κ,μ)-manifolds

2005 ◽  
Vol 2005 (22) ◽  
pp. 3575-3586 ◽  
Author(s):  
K. Arslan ◽  
R. Ezentas ◽  
C. Murathan ◽  
T. Sasahara
Keyword(s):  
Riemannian Manifolds ◽  
Critical Points ◽  
Harmonic Maps ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Biharmonic Maps ◽  
Biharmonic Submanifolds ◽  
Invariant Surfaces ◽  
Legendre Curves ◽  
Necessary And Sufficient

Biharmonic maps between Riemannian manifolds are defined as critical points of the bienergy and generalized harmonic maps. In this paper, we give necessary and sufficient conditions for nonharmonic Legendre curves and anti-invariant surfaces of3-dimensional(κ,μ)-manifolds to be biharmonic.

scholarly journals Deformations of Metrics and Biharmonic Maps

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Aicha Benkartab ◽  
Ahmed Mohammed Cherif
Keyword(s):  
Riemannian Manifolds ◽  
Smooth Function ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Biharmonic Maps

AbstractWe construct biharmonic non-harmonic maps between Riemannian manifolds (M, g) and (N, h) by first making the ansatz that φ : (M, g) → (N, h) be a harmonic map and then deforming the metric on N by{\tilde h_\alpha } = \alpha h + \left( {1 - \alpha } \right){\rm{d}}f \otimes {\rm{d}}fto render φ biharmonic, where f is a smooth function with gradient of constant norm on (N, h) and α ∈ (0, 1). We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

2002 ◽  
Vol 04 (04) ◽  
pp. 725-750 ◽  
Author(s):  
CHIKAKO MESE
Keyword(s):  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Harmonic Maps ◽  
Uniqueness Theorems ◽  
Singular Spaces ◽  
Domain Space ◽  
Recent Developments

Recent developments extend much of the known theory of classical harmonic maps between smooth Riemannian manifolds to the case when the target is a metric space of curvature bounded from above. In particular, the existence and regularity theorems for harmonic maps into these singular spaces have been successfully generalized. Furthermore, the uniqueness of harmonic maps is known when the domain has a boundary (with a smallness of image condition if the target curvature is bounded from above by a positive number). In this paper, we will address the question of uniqueness when the domain space is without a boundary in two cases: one, when the curvature of the target is strictly negative and two, for a map between surfaces with nonpositive target curvature.

The convex hull of a spherically symmetric sample

1981 ◽  
Vol 13 (4) ◽  
pp. 751-763 ◽  
Author(s):  
William F. Eddy ◽  
James D. Gale
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Probability Measure ◽  
Convex Hull ◽  
Random Sample ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Extreme Value ◽  
Spherically Symmetric ◽  
Extreme Value Distributions

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

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