Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

A study on killing vector fields in four-dimensional spaces

In the present study, some properties of Killing vector fields are investigated on 4-dimensional manifolds in case of the signature of the metric tensor 1 is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.

Some invariants of conformal mappings of a generalized Riemannian space

Invariants of conformal mappings between non-symmetric affine connection spaces are obtained in this paper. Correlations between these invariants and the Weyl conformal curvature tensor are established. Before these invariants, it is obtained a necessary and sufficient condition for a mapping to be conformal. Some appurtenant invariants of conformal mappings are obtained.

On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds

In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.

CONFORMALLY OSSERMAN MANIFOLDS AND CONFORMALLY COMPLEX SPACE FORMS

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

IMPROVED FORMS OF SOME VANISHING THEOREMS IN RIEMANNIAN SPIN GEOMETRY

We improve the hypotheses on some vanishing theorems for first order differential operators on bundles over a Riemannian spin manifold. The improved hypotheses are uniform, in the sense that they are the same for each of an infinite sequence of bundles in each even dimension. They are also elementary, in the sense that they involve only the bottom eigenvalue of the Yamabe operator on scalars, and the pointwise action of the Weyl conformal curvature tensor on two-forms. In particular, they do not make reference to the higher spin bundles on which the conclusion holds.

Some solutions of the Lanczos vacuum wave equation

In general relativity the non-local part of the gravitational field is described by the 10 degrees of freedom of the Weyl conformal curvature tensor C abcd . In every space-time the Weyl field C abcd is derivable from a potential L abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions L ab s ; s = 0 are imposed. The potential L abc discov­ered by Lanczos was shown by Illge to have an extremely simple vacuum wave equation, namely, □ L abc ≡ g sm L abc ; s ; m = 0. Using tensor, spinor and spin-coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields.