Umbilical indicatrices of the unit tangent vector and the principal normal and binormal vectors of a curve in Riemannian space

1956 ◽  
Vol 44 (5) ◽  
pp. 351-359
Author(s):  
H. S. Nagarathnamma
Keyword(s):  
Riemannian Space ◽  
Tangent Vector ◽  
Unit Tangent Vector

On a Generalization of the First Curvature of a Curve in a Hypersurface of a Riemannian Space

1954 ◽  
Vol 6 ◽  
pp. 210-216 ◽  
Author(s):  
T. K. Pan
Keyword(s):  
Riemannian Space ◽  
Tangent Vector ◽  
Curvature Vector ◽  
Tangential Component ◽  
Unit Tangent Vector

The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.

Note on unit tangent vector computation for homotopy curve tracking on a hypercube

1991 ◽  
Vol 17 (12) ◽  
pp. 1385-1395 ◽  
Author(s):  
A. Chakraborty ◽  
D.C.S. Allison ◽  
C.J. Ribbens ◽  
L.T. Watson
Keyword(s):  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Curve Tracking

scholarly journals SYMMETRIC TELEPARALLEL GRAVITY: SOME EXACT SOLUTIONS AND SPINOR COUPLINGS

2013 ◽  
Vol 28 (32) ◽  
pp. 1350167 ◽  
Author(s):  
MUZAFFER ADAK ◽  
ÖZCAN SERT ◽  
MESTAN KALAY ◽  
MURAT SARI
Keyword(s):  
Riemannian Space ◽  
Tangent Vector ◽  
Teleparallel Gravity ◽  
Coupling Coefficients ◽  
Field Equations ◽  
Minimal Coupling ◽  
Zero Curvature ◽  
Pp Wave ◽  
Teleparallel Theory ◽  
Parallel Transportation

In this paper, we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian space–time with nonzero nonmetricity, but zero torsion and zero curvature. First, we give a prescription for obtaining the nonmetricity from the metric in a peculiar gauge. Then, we state that under a novel prescription of parallel transportation of a tangent vector in this non-Riemannian geometry, the autoparallel curves coincide with those of the Riemannian space–times. Subsequently, we represent the symmetric teleparallel theory of gravity by the most general quadratic and parity conserving Lagrangian with lagrange multipliers for vanishing torsion and curvature. We show that our Lagrangian is equivalent to the Einstein–Hilbert Lagrangian for certain values of coupling coefficients. Thus, we arrive at calculating the field equations via independent variations. Then, we obtain in turn conformal, spherically symmetric static, cosmological and pp-wave solutions exactly. Finally, we discuss a minimal coupling of a spin-1/2 field to STPG.

Bounded geodesics and Hausdorff dimension

1994 ◽  
Vol 116 (3) ◽  
pp. 505-511 ◽  
Author(s):  
C. S. Aravinda
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Tangent Vector ◽  
Unit Tangent Vector ◽  
Constant Negative Curvature ◽  
Compact Closure ◽  
Riemannian Volume ◽  
Unit Tangent Bundle ◽  
Almost All

Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all (p, v)∈ SM, where p ∈M and v is a unit tangent vector at p, the geodesic through p in the direction of v is dense in M. A theorem of Dani [Dl] says that the set of all (p, v)∈SM for which the corresponding geodesic is bounded (namely those with compact closure in M) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

scholarly journals On the concept of gravitational force and Gauss's theorem in general relativity

1937 ◽  
Vol 5 (2) ◽  
pp. 93-102 ◽  
Author(s):  
J. L. Synge ◽  
H. S. Ruse
Keyword(s):  
General Relativity ◽  
Physical Meaning ◽  
Gravitational Force ◽  
Normal Force ◽  
Free Particle ◽  
Tangent Vector ◽  
Curvature Vector ◽  
Unit Tangent Vector ◽  
Normal Congruence ◽  
Definition Of

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.

scholarly journals Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1018
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Differential Equations ◽  
Riemannian Space ◽  
Dimensional Subspace ◽  
Geometric Construction ◽  
Lie Point Symmetries ◽  
Geodesic Equations ◽  
Symmetries Of Differential Equations ◽  
Projective Collineations

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

scholarly journals Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

2021 ◽  
Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Riemannian Space ◽  
Space Forms ◽  
Type Formula ◽  
Isometric Embeddings ◽  
Curvature Measures ◽  
Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

Classical mechanics of special relativity in a Riemannian space‐time

1991 ◽  
Vol 32 (7) ◽  
pp. 1788-1795 ◽  
Author(s):  
Daniel Zerzion ◽  
L. P. Horwitz ◽  
R. I. Arshansky
Keyword(s):  
Special Relativity ◽  
Riemannian Space ◽  
Classical Mechanics ◽  
Space Time
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