Holomorphically proyective Killing fi elds with vectorial fi elds associated in kahlerian manifolds

Taking into account the harmonic and scalar curvatures in the study of Killing transformations between spacial complex (Einstenian, Peterson-Codazzi, Recurrent) and kaehlerian M spaces with almost complex J structure, we prove that there exists an holomorphically proyective transformation between M spaces and complex spaces.


Introduction and preliminaries
By the end of the 20 th century researchers started to link the concept of proyectivity with the phenomenon of complex manifolds specially in terms of their hollomorphic properties.Then, kaehlerian and hermithian manifolds as well as complex hyper surfaces and other manifolds were considered embedded into special transformations.At this point a vast number of publications arose in relation with the concepts of compact K manifolds, proyective infinitesimal transformations in Riemmanian manifolds with additive curvature properties and hollomorphic proyective equivalences and others.Based on [1] [2] and [3], this research studies Kaehler holomorphically proyective manifolds with almost complex structures by using the geometric properties of the harmonic and scalar curvatures evaluated over Killing vectorial fields.Two important applications result from this, the Einstenian and the constant curvature spaces.
Considering (M, g, J) as a kaehlerian manifold of 2n ≥ 4 dimension with a g = (g ij ) Riemannian metric and an almost-complex structure J = J ij where J ij = −J ji and with a Riemannian curvature tensor of aji then the Ricci tensor and the r = g ba S ba scalar curvature satisfy the following proprieties: where The Lie operator derivative in the vectorial field direction X for R h kji and hji is represented respectively by, ii) X is an holomorphically projective transformation when is a particular vector associated to X. Two metric g = (g ij ) y g = (g ij ) defined on M , they are hollomorphic proyective equivalences if where e F i = J a i F a .Tensors for harmonic and scalar curvatures are defined on the manifold M by means of the following relations: respectively where and F j = ∇ j f .The classic commutative relationship of L X and de ∇ for a tensor Y of (1,2) type is given by Being X an holomorphically proyective transformation with an F associated vector then the following identities are satisfied, watch [1] i In [3] proof ijk it is a recurrent space where F l 6 = 0 or it is an Einstein space if S = λg taking S as the Ricci and g an the metric tensors and λ as a parameter.
Lemma 1.1.If M is a compact Kaehlerian manifold of dimension n with a scalar curvature R and it admits an holomorphically proyective transformation then the following equations are fulfilled, Proof.i) Since A n is a recurrent space and M admits an holomorphically proyective transformation then we obtain multiplying (1.4) by g hk and applying ∇ b it results that Now using Ricci 0 s and Bianchi 0 s identities we obtain finally by applying ∇ j the result is ii) The demonstration is obtained by using I and part (i) from this lema.
Lemma 1.2.Let X be an holomorphically proyective transformation with an F associated vector then Holomorphically proyective Killing fields with vectorial fields ... 223 Proof.Using the definition of L X R h kji we have since X is an holomorphically projective transformation then By considering the real part we obtain the desired result

Results
The following theorem allows a Kaehlerian space to become into a Peterson-Codazzi space under the hypothesis that the former is holomorphically projective.
Theorem 2.1.Let M be a Kaehlerian manifold and X be an holomorphically projective killing field with an associated vectorial field F then Proof.Using the classic relation of commutation for a (0,2) type tensor we obtain that If by hypothesis we consider X as an holomorphically projective transformation by using (1.2) then we have that and analogically we obtain L X Γ a ki and L X S ki .By Substituting (2.3) and (2.4) By doing certain manipulations and using simplification we conclude that n From here on some applications of the previous results will be given. (2.5)

Consequence i
A Kaehler-Peterson-Codazzi space has an harmonic curvature since, Consequence ii A Kaehler-Peterson-Codazzi space is an Einstenian space if the former has a constant scalar curvature.Factually by applying g ki into (2.5)results in Since F a 6 = 0 and developing the three last terms we have, by making the contraction a = j and adding up from 1 to n we obtain In this way we conclude that S ki = R n g ki .In other words the Kaehler-Peterson-Codazzi space is an Einstenian space.
2) If M is a recurrent space then

Consequence
If M is an harmonic curvature and then M has a null scalar curvature.
As a matter of fact if M admits an harmonic curvature then making the contraction l = a and summing up from 1 to n in the relation and multiplying the previous relation by g ki it results that wherein by applying F j it results that r = 0.This way we conclude that the manifold is plain.

Example
Be Einstein compact Kaehlerian spaces A n = (M, ∇) and A n = (M, ∇), with metric g = (g ij ) y g = (g ij ) hollomorphic proyective equivalences, get an expression that relates the scalar curvature R and R.