ISRAEL JOURNAL OF MATHEIVlATICS,Vol. 53, No, 3, 1986
Ax-OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS
BY CARLOS CURRAS-BOS...
41 downloads
455 Views
203KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
ISRAEL JOURNAL OF MATHEIVlATICS,Vol. 53, No, 3, 1986
Ax-OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS
BY CARLOS CURRAS-BOSCH
Dpto. Geometr[a y Topologth, Facultad Matemdticas, Universidad de Barcelona, Barcelona, Spain
ABSTRACT
In this paper we give a generalisation of Kostant's Theorem about the Ax-operator associated to a Killing vector field X on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operator A× = L x - V x associated to a Killing vector field X lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M.g) the A×-operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-fiat manifolds such that Ax does not lie in the holonomy algebra at any point. §1.
Preliminary results
Let ( M , g )
be a c o m p l e t e n o n - c o m p a c t
Riemannian
manifold. O n e can
d e c o m p o s e the Lie algebra of s k e w - s y m m e t r i c e n d o m o r p h i s m s x E M in the form at x and
E(x) = G(x)O G(x) l,
G(x) ± its
where
G(x)
E(x)
at a point
is the h o l o n o m y algebra
o r t h o g o n a l c o m p l e m e n t with respect to the local scalar
p r o d u c t for tensors of type (1, 1). Since xGM
Ax
is s k e w - s y m m e t r i c for each Killing vector field X, then at any point
we have:
(Ax),=(Sx)~+(Bx),
with(S×)x~G(x)
and
(Bx),EG(x) I.
Notice that if B× = 0 at s o m e point, the B× vanishes e v e r y w h e r e , because Bx is parallel. For two tensor fields T, S of type (0, s) on M, we d e n o t e the local scalar p r o d u c t of T and S by (T, S), i.e.
( T , S ) = I z,._,S', '', Received February 27, 1984 315
316
C. CURR/~S-BOSCH
Isr. J. Math.
and the global scalar product by ((T, S)),
((T,S)) = fM (r, S)vol. We denote by A ' ( M ) (resp. A~(M)) the space of s-forms on M (resp. with compact support). Let L~(M) be the completion of M ( M ) with respect to (( , )). We say that a vector field X has a finite global norm (see [8]), if the 1-form associated to X via g, ~:, lies in L~(M)A A'(M). To prove our result we have to use Stokes' Theorem and to do this we construct a family of cut-ott functions whose supports exhaust M (see [t], [4] or [8]). Let 0 be a fixed point of M. For each p E M we denote by p(p) the geodesic distance from 0 to p. Let B ( a ) = { p E M I p ( p ) < a } , 4 > 0 . We choose a C~-function/z on R satisfying:
(i) 0_-