Submanifolds and the second fundamental tensor

1985 ◽  
pp. 194-203 ◽  
Author(s):  
Stefana Hineva
Keyword(s):  
Fundamental Tensor

Generalized Riemann spaces

1956 ◽  
Vol 52 (4) ◽  
pp. 623-625 ◽  
Author(s):  
John Moffat
Keyword(s):  
Curvature Tensor ◽  
Physical Theory ◽  
Dimensional Space ◽  
Riemann Space ◽  
Recent Attempt ◽  
Fundamental Tensor ◽  
Fundamental Properties ◽  
Complex Symmetric ◽  
Definition Of ◽  
Generalized Curvature Tensor

ABSTRACTThe recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

scholarly journals On the Stability in Bonnet's Theorem of the Surface Theory

2007 ◽  
Vol 14 (3) ◽  
pp. 543-564
Author(s):  
Yuri G. Reshetnyak
Keyword(s):  
Differential Operators ◽  
Sobolev Class ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Integral Representations ◽  
Completely Integrable Systems ◽  
Frobenius Theorem ◽  
Fundamental Tensor ◽  
The Stability ◽  
Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

scholarly journals On real hypersurfaces in quaternionic projective space with𝒟⊥-recurrent second fundamental tensor

1999 ◽  
Vol 22 (1) ◽  
pp. 109-117
Author(s):  
Young Jin Suh ◽  
Juan De Dios Pérez
Keyword(s):  
Projective Space ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Quaternionic Projective Space ◽  
Fundamental Tensor ◽  
Orthogonal Distribution

In this paper, we give a complete classification of real hypersurfaces in a quaternionic projective spaceQPmwith𝒟⊥-recurrent second fundamental tensor under certain condition on the orthogonal distribution𝒟.

scholarly journals Submanifolds and the length of the second fundamental tensor

1983 ◽  
Vol 34 (1) ◽  
pp. 1-6
Author(s):  
Thomas Hasanis
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Mean Curvature ◽  
Sufficient Condition ◽  
Fundamental Tensor ◽  
The Mean

AbstractA sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

XI.—Electromagnetic Phenomena in a Uniform Gravitational Field

1932 ◽  
Vol 51 ◽  
pp. 71-79 ◽  
Author(s):  
D. Meksyn
Keyword(s):  
Electromagnetic Field ◽  
Gravitational Field ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Suitable Choice ◽  
Fundamental Tensor

In two recent papers Professor E. T. Whittaker has solved the electromagnetic equations for the case of a uniform gravitational field. The fundamental tensor associated with such a field makes the Riemannian tensor vanish, since such a field can be transformed away by a suitable choice of coordinates. This property enables us to find the electromagnetic field in a uniform gravitational field without solving Maxwell's equations, but by a mere transformation of co-ordinates.

scholarly journals Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Nikolay Yaremenko
Keyword(s):  
Curvature Tensor ◽  
Geometric Structure ◽  
Jacobi Identity ◽  
Torsion Tensor ◽  
Variation Principle ◽  
Field Equations ◽  
Metric Tensor ◽  
Fundamental Tensor ◽  
Geodesic Lines ◽  
General Field

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).

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