SECOND ORDER PARALLEL TENSOR IN REAL AND COMPLEX SPACE FORMS

Levy’s theorem "A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor-" has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form. it has been shown that an affine Killing vector field in a non-flat complex space form is Killing and analytic.


INTRODUCTION.
In 1923, Eisenhart [l] proved that if a positive definite Riemannian manifold admits a second order-parallel symmetric tensor other than a constant multiple of the metric tensor, then it is reducible. ]n 1926, Levy [2] proved that a second order parallel symmetric non-singular (with non-vanishing determinant) tensor in a space of constant curvature is proportional to the metric tensor.
of this pape..r is to present a eneralization over Levy's theorem for dimension greater than two in the form .pf Theorem and its an.a__lqg.._e in a Kaehlerian manifold of constant holomorphic sectional curvature also called a comp!ex space form) in the f_q_rm of Theorem 2. Using_ Theorem__2_ it has been proved in Theorem 3 t_h_a_t___.anaffine Killing vector field in a non-flat co,nplex space form is Killing and analytic.
Let M denote an n-dimensional pseudo-Riemannian manifold with its metric tensor g of arbitrary signature and Levi-Civita connection v. Let R denote the Riemann curvature tensor of M. If h is a (0,2)-tensor which is parallel with respect to v then we can show easily that h(R(X,Y)Z,W) + h(Z,R(X,Y)W) 0 (l.l) 2. A GENERALIZATION OF LEVY'S 'rIIEOREM.
, 4 second order parallel tensor in a non-flat real space form dimet,qi,r r 2 i; prportional to the melric tcnsor.

PROOF:
For a real space form M with constant sectional ctrvature k, we hay(, Note that k ; 0, by hypothesis.Use of (2. l) in (l.1) gives Contraction at X ad W with respect to an orthonormal frame in M, provides where H is a (l,1)-tensor metrically equivalent to h.Anti-symmetrization of (2.3) shows that h is symmetri(:.Eventually (2. (see the beginning of section 3) whose Kaehlerian 2-form is a parallel tensor.

ANALOGUF, OF THEOREM
FOR A COMPI,EX SPACIg FORM.
Before presenting an analogue of Theorem for a complex space-form, we would like to recall the basic structure of a complex space form M(c). M((:) is a Kaehlerian manifold of constant holomorphic sectional curvature c, with its complex structure tensor J J -l, gaeh]erian metric g g(JX,JY) g(X,Y), Kaehlerian 2-form fl i(X,Y) = g(X,JY) and the gaehlerian connection v vJ 0. THRORRM 2. A second order parallel tensor in a non-flat complex space form is a linear combination (with constant coefficients) of the underlying Kaehlerian metric and Kaehlerian 2-form.
PROOF: For a complex space form M(c), it is known 13] that C R(X,Y)Z [g(Y,Z)Xg(X,Z)Y g(JY,Z)JX-g(J)l,g)JY + 2g(X,JV)Jg} Plugging the value ()f 14 from (3.1) into (l.1) and contracting at X and W, (3.7) n Now as both H and J are parallel with respect to v; therefore tr.H and tr.HJ are constants.Thus, Equation {3.7) proves the theorem.
COROLLARY.The only symmetric (anti-symmetric) parallel tensor of type (0,2) in a non-flat complex space form is the Kaehlerian metric (the  up to , constant multiple.
REMARK 2. The anti-symmetric case of the above corollary agrees well with the following result [3]: "In a compact Kaehlerian space of constant holomorphic sectional curvature c > 0, we have B 1, B+, 0 for 0 25, 25 + n)'.
Taking $ 1, the second Betti number B for a compact M(c) with c 0.
Thus the only harmonic 2-form in such a space is the Kaehlerian 2-form fl (Note that vfl : 0 implies dr} : 0 and 6fl : 0, that is, fl is harmonic).THEOREM 3.An affine Killing vector field in a non-flat complex space form is Killing and analytic.
PROOF: If is an affine Killing vector field in a non-flat M(c), then the Lie-derivative Lg of the metric tensor g is a second order parallel tensor.A direct application of the symmetric case of the corollary to Theorem 2, shows that Lg : ag (a being a constant).The last equation implies that LRic : 0 (Ric denotes the Ricci tensor of M(c)).Now, we know [3] that M(c) is an Einstein n+2 space, that is, Ric 4 cg.Taking lhe Lie-derivatives of both sides along and noting c / 0, obtain Lg 0. Hence is Killing.
In Theorem 3 we have proved that a Killing vector field in a non-flat complex space form is analytic vector field of J.One can compare this result with the following result of Yano [3]: "A Killing vector field in a compact Kaehler space is analytic'.Our result assumes the vector field to be just affine Killing and proves it to be Killing and analytic in a complex space form (not necessarily compact), whereas Yano's result proves a Killing vector to be analytic if the space is compact Kaehler (not necessarily of constant holomorphic sectional curvature).