Differential geometry on complex and almost complex spaces

In the present paper we continue to study almost geodesic curves and determine in 
$${\mathbb {R}}^n$$

 the form of curves 
$${\mathcal {C}}$$

 for which every image under an 
$$(n-1)$$

-dimensional algebraic torus is also an almost geodesic with respect to an affine connection 
$$\nabla $$

 with constant coefficients. We also calculate explicitly the components of 
$$\nabla $$

. For the explicit calculation of the form of curves 
$${\mathcal {C}}$$

 in the n-dimensional real space 
$${\mathbb {R}}^n$$

 that are almost geodesics with respect to an affine connection 
$$\nabla $$

, we can suppose that with 
$${{\mathcal {C}}}$$

 all images of 
$${{\mathcal {C}}}$$

 under a real 
$$(n-1)$$

-dimensional algebraic torus are also almost geodesics. This implies that the determination of 
$${{\mathcal {C}}}$$

 becomes an algebraic problem. Here we use E. Beltrami’s result that a differentiable curve is a local geodesic with respect to an affine connection 
$$\nabla $$

 precisely if it is a solution of an abelian differential equation with coefficients that are functions of the components of 
$$\nabla $$

. Now we consider the special case for the connection 
$$\nabla $$

 in which every curve is almost geodesic with respect to 
$$\nabla $$

.

Almost Geodesics and Special Affine Connection

Periodic J-trajectories on R×S3

In the present paper we obtain a new proof of the fundamental equations of F-planar mappings of manifolds with affine connections. We discuss alternative ways in the definition of F-planar mappings.

Fundamental equations of F-planar mappings

Let $${(W, q, \mathcal{D})}$$
 be a Walker manifold. We find all Walker metrics which are harmonic [in the sense of Chen and Nagano in (J Math Soc Jpn 36:295–313, 1984)] w.r.t. q. On the total space of the tangent bundle of W, we obtain necessary and sufficient conditions concerning the harmonicity of certain metrics w.r.t. the Sasaki (resp. horizontal, vertical) lift of q. The harmonicity in the sense of Garcia-Rio et al. in (Ill J Math 41(1):23–30, 1997) of the three endomorphism fields of an almost hyper-para-Hermitian structure is characterized. As an application, we deal with mixed 3-structures, a notion for which we quote Ianus in (Mediterr J Math 3(3–4):581–592, 2006).

Structures Which are Harmonic with Respect to Walker Metrics

The notion of F-geodesic, which is slightly different from that of F-planar curve (see [14], [15]), generalizes the magnetic curves, and implicitly the geodesics, by using any (1,1)-tensor field on the manifold (in particular the electro-magnetic field or the Lorentz force). We give several examples of F-geodesics and the characterizations of the F-geodesics w.r.t. Vranceanu connections on foliated manifolds and adapted connections on almost contact manifolds. We generalize the classical projective transformation, holomorphic-projective transformation and C-projective transformation, by considering a pair of symmetric connections which have the same F-geodesics. We deal with the transformation between such two connections, called F-projective transformation. We obtain a Weyl type tensor field, invariant under any F-projective transformation, on a 1-codimensional foliation.

/pdf/f-geodesics-on-manifolds-5dlcwsquhw.pdf

F-geodesics on manifolds

Let (W, q,D) be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special (1, 1)-tensor fields on (W, q,D) , we prove that the proper almost complex structure J , introduced by Matsushita, is harmonic in the sense of Garcia-Ŕio et al. if and only if the almost Hermitian structure (J, q) is almost Kahler. We classify all harmonic functions locally defined on (W, q,D) . We deal with the harmonicity of quadratic maps defined on R (endowed with a Walker metric q ) to the n -dimensional semi-Euclidean space of index r , and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to q .

/pdf/harmonic-functions-and-quadratic-harmonic-morphisms-on-2ifpp8whr5.pdf

Simona-Luiza Druta-Romaniuc

Papers

F-geodesics on manifolds

Harmonic functions and quadratic harmonic morphisms on Walker spaces