New Developments in Differential Geometry pp 335-342 | Cite as

# A differential equation related with some general connections

## Abstract

A general connection *Γ* on a differentiable manifold *M* ^{ n } is given as a geometrical object with components (P^{i} _{j}T^{i} _{j}h) in local coordinates (u^{i}) such that *(P* ^{ i } _{ j } *)* are the components of a tensor and (T^{i} _{j}) satisfy the rules : Γ< _ dv’ f p k g V M0vT\ *Jh ~ duk \ 1 dvidvh + lmdvi dvh J ’* where *T* ^{ i } _{ j } are the second components of *Γ* in local coordinates (v^{i}). We call a point of *M* ^{ n } is regular or singular with respect to *Γ* if det *(P* ^{ i } _{ j } *)* ≠ 0 or = 0, respectively. We say a curve *γ(t) = (u* ^{ i } *(t))* is a geodesic with respect to *Γ* if *u* ^{ i } *(t)* satisfy the equations: *ni d2v? „, du* duh „* and the parameter *t* is called its affine parameter.

## Keywords

Scalar Curvature Galerkin Method Hyperbolic Space Ricci Tensor Nonlinear Partial Differential Equation## Preview

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## References

- 1.T. Otsuki: General connections,
*Math. J. Okayama Univ*. 32(1990), 227–242.MathSciNetzbMATHGoogle Scholar - 2.T. Otsuki: Behavior of geodesies around the singular set of a general connection,
*SUT Journal of Math*. 27(1991), 169–228.MathSciNetzbMATHGoogle Scholar - 3.T. Otsuki: A family of Minkowski-type spaces with general connections,
*SUT J. Math*. 28 (1992), 61–103.MathSciNetzbMATHGoogle Scholar - 4.T. Otsuki: A nonlinear partial differential equation related with certain spaces with general connections,
*SUT J. Math*. 29 (1993), 167–192.MathSciNetzbMATHGoogle Scholar