scholarly journals Autoparallel Deviation in the Geometry of Lyra

1960 ◽  
Vol 3 (3) ◽  
pp. 263-271 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Calculus Of Variations ◽  
Finsler Manifold ◽  
Geometrical Approach ◽  
Second Variation ◽  
Geodesic Deviation ◽  
The Difference ◽  
The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

Complete Riemannian Manifold with Curvature Bounded from Below

2012 ◽  
Vol 182-183 ◽  
pp. 1225-1229
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao
Keyword(s):  
Riemannian Manifold ◽  
Complete Riemannian Manifold ◽  
Second Variation ◽  
Variation Formula ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
First Variation ◽  
Geodesic Submanifold ◽  
The First Variation ◽  
The Second Variation

In this paper, we study a complete -Riemannian manifold whose curvature bounded from below. Let be a compact totally geodesic submanifold of . Then, for any , we can make use of the first variation formula and the second variation formula of distance to prove that is bounded.

scholarly journals III. On the discrimination of maxima and minima solutions in the calculus of variations

1886 ◽  
Vol 40 (242-245) ◽  
pp. 476-477
Keyword(s):  
Calculus Of Variations ◽  
Second Variation ◽  
Additional Advantage ◽  
Differential Coefficient ◽  
Undetermined Function ◽  
Independent Variable ◽  
The Second Variation

In the first part of the paper it is shown that the usual investigation by which the second variation of an integral is reduced, requires that the variation given to y (the undetermined function) is such that its differential coefficients, taken with regard to x (the independent variable) are continuous up to the twice- n th order, d n y / dx n being the highest differential coefficient of y appearing in the function to be integrated. But it is not necessary that the variation should be continuous beyond its ( n —1)th differential coefficient, and a method of reducing the variation to Jacobi’s form by a process which is not open to the above objection is then given; and the method has the additional advantage that its simplicity enables it to be easily extended to other cases where there are more than two variables.

On a gauge-invariant functional

2010 ◽  
Vol 53 (1) ◽  
pp. 143-151
Author(s):  
Cătălin Gherghe
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Compact Riemannian Manifold ◽  
Second Variation ◽  
Gauge Invariant ◽  
Variation Formula ◽  
First Variation ◽  
Yang Mills ◽  
Invariant Functional ◽  
The Second Variation

AbstractWe define a new functional which is gauge invariant on the space of all smooth connections of a vector bundle over a compact Riemannian manifold. This functional is a generalization of the classical Yang-Mills functional. We derive its first variation formula and prove the existence of critical points. We also obtain the second variation formula.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals A Note On Umbilics of a Closed Surface

1959 ◽  
Vol 15 ◽  
pp. 219-223
Author(s):  
Minoru Kurita
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Closed Surface ◽  
Dimension 2 ◽  
Direction Field

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

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