In this note we outline a proof of the Calderon extension theorem by a technique similar to that for the Whitney extension theorem. For classical proofs, see Calderon [2] and Morrey [4]. See also Palais [6, p. 170]. Our purpose is thus to give a more unified proof of the theorem in the various cases. In addition, the proof ap.plies to the Holder spaces C+, which was used in [3], and applies to ...

How many are projectable classical linear connections with a prescribed Ricci tensor and trace of torsion on the total space fibered manifold? The questions answered in analytic case by using Cauchy-Kowalevski theorem. In C? case, we answer how 2-dimensional manifold. also deduce that any 2-form manifold at least fibres can be realized locally as connection.

For a $$2n+1$$ -dimensional compact Sasakian manifold, if $$n\ge 2$$ , we prove that the analytic germ of variety representations fundamental group at every semi-simple representation is quadratic. To this result, almost-formality de Rham complex manifold with values in flat vector bundle. By almost-formality, also vanishing theorem on cup product cohomology bundles over manifold.

We prove a vanishing theorem for a convex cocompact hyperbolic manifold, which relates its L cohomology and the Hausdorff dimension of its limit set. The borderline case is shown to characterize the manifold completely.

We prove a vanishing theorem for a convex cocompact hyperbolic manifold which relates its L2-cohomology and the Hausdorff dimension of its limit set. The borderline case is shown to characterize the manifold completely.

Let M be an arbitrary Riemannian manifold diffeomorphic to S2. Let x, y be two arbitrary points of M. We prove that for every k = 1, 2, 3, . . . there exist k distinct geodesics between x and y of length less than or equal to (4k2 − 2k − 1)d, where d denotes the diameter of M. To prove this result we demonstrate that for every Riemannian metric on S2 there are two (not mutually exclusive) possi...

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts I, II. In Part I (this pap...

The object of this paper is to define and study a new type of non-flat Riemannian manifold called nearly Einstein manifold. The notion of this nearly Einstein manifold has been established by an example and an existence theorem. Some geometric properties are obtained. AMS Mathematics Subject Classification (2010): 53C25.

Recall that the radius of a compact metric space (X, dist) is given by rad X = minx∈X maxy∈X dist(x, y). In this paper we generalize Berger’s 1 4 -pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature ≥ 1 and radius ≥ π2 is either homeomorphic to the sphere or isometric to a compact rank one symmetric space. The classical sphere theorem ...

In this article, we first consider the L2 Morse–Novikov cohomology on a complete Riemannian manifold M equipped with parallel 1-form which includes Vaisman manifold. Based vanishing theorem of cohomology, prove that L2-harmonic forms are identically zero.