In the present paper, we study geometry of infinitesimal conformal, affine, projective, and harmonic transformations complete Riemannian manifolds using concepts geometric dynamics methods modern version Bochner technique.

I J. P. Serre, "Fonctions automorphes," &minaire 1cole Norm. Sup., 2, (Paris 1953/4); C. L. Siegel, "Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten," Nachr, A kad. Wiss. Gottingen, 2a, 71-77, 1955. R. Remmert, 'Meromorphe Funktionen in kompakten komplexen Rdumen," Math. Ann., 132, 277-288, 1956. 5S. Bochner, and W. T. Martin, 'Complex Spaces with Singularities," Ann. Math.,...

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

Let E be a separable Banach space and Ω be a compact Hausdorff space. It is shown that the space C(Ω, E) has property (V) if and only if E does. Similar result is also given for Bochner spaces L(μ,E) if 1 < p < ∞ and μ is a finite Borel measure on Ω.

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing...

We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.

We study closed Einstein 4-manifolds which admit S1 actions of a certain type, i.e., warped products. In particular, we classify them up to isometry when the fixed point of the S1 action satisfies certain natural geometric conditions. The proof uses the Bochner-Weitzenböck formula for 1-forms and the theory of minimal surfaces in 3-manifolds.

We study the lattice finite representability of the Bochner space Lp(μ1, Lq(μ2)) in `p{`q}, 1 ≤ p, q < ∞, and then we characterize the ideal of the operators which factor through a lattice homomorphism between L∞(μ) and Lp(μ1, Lq(μ2)).