Multidimensional parametrized variational problems on riemannian manifolds

1986 ◽  
pp. 40-62
Author(s):  
Đào Trong Thi
Keyword(s):  
Riemannian Manifolds ◽  
Variational Problems

scholarly journals Riemannian Means as Solutions of Variational Problems

2006 ◽  
Vol 9 ◽  
pp. 86-103 ◽  
Author(s):  
Luís Machado ◽  
F. Silva Leite ◽  
Knut Hüper
Keyword(s):  
Riemannian Manifold ◽  
Variational Problem ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Single Point ◽  
Variational Problems ◽  
Compact Lie Groups ◽  
Data Set ◽  
The Given ◽  
Best Fit

We formulate a variational problem on a Riemannian manifoldMwhose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points inM. By a limiting process, these solutions converge to a single point inM. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR

2012 ◽  
Vol 09 (04) ◽  
pp. 1250032 ◽  
Author(s):  
IMSOON JEONG ◽  
HYUNJIN LEE ◽  
YOUNG JIN SUH
Keyword(s):  
Lie Algebras ◽  
Riemannian Manifolds ◽  
Shape Operator ◽  
Variational Problems ◽  
Real Hypersurfaces ◽  
Graded Lie Algebras ◽  
Totally Geodesic ◽  
Cartan Connections ◽  
Parallel Shape Operator ◽  
Kodai Math

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G2(ℂm+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇⊥-parallel for a hypersurface M in G2(ℂm+2), and give a characterization for a tube around a totally geodesic ℍ Pn in G2(ℂm+2) where m = 2n.

scholarly journals Efficiency for Vector Variational Quotient Problems with Curvilinear Integrals on Riemannian Manifolds via Geodesic Quasiinvexity

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1054
Author(s):  
Tiziana Ciano ◽  
Massimiliano Ferrara ◽  
Ştefan Mititelu ◽  
Bruno Antonio Pansera
Keyword(s):  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Invex Set ◽  
Fractional Variational Problems ◽  
Efficiency Conditions

In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( ρ , b)-geodesic quasiinvex functions.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

2019 ◽  
Vol 16 (4) ◽  
pp. 557-566
Author(s):  
Denis Ilyutko ◽  
Evgenii Sevost'yanov
Keyword(s):  
Riemannian Manifolds ◽  
Type Inequality ◽  
The Given ◽  
Range Of Values

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

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