The problem of finding a george joinning given points x0, x1 in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available expectually when x0, x1 are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case.

Our global algorithm is very simple to implement, and works well in practice, with no need for an initial guess. The proof of convergence to elementary and very carefully stated. with a view to possible generalizations latter on we have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering.

Let a compact orientable manifold be immersed as a hypersurface of constant mean curvature in an Einstein space. It is shown that the immersion is totally umbilical if and only if there exists a conformal variation of the immersion whose variation vector is nowhere tangential to the hypersurface.

We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

We consider hypersurfaces of En+1 whose position vector x satisfies Δx = Ax + B, where Δ is the induced Laplacian, and prove that these are open parts of minimal hypersurfaces, hyperspheres or generalized circular cylinders.

We introduce a new homology theory for infinite graphs in order to generalize some results of Willis and Woodward on translation invariant functionals. We also extend some theorems of Gerl and Gromov.

In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

We study real hypersurfaces of a complex projection space and show that there are no such hypersurfaces with harmonic curvature on which the structure vector is principal.

Let N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.

A sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.

We show that a warped product Mf = nf has higher rank and nonpositive curvature if and only if f is a convex solution of the Monge-Ampère equation. In this case we show that M contains a Euclidean factor.

For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

Defining a function on the set of all Riemannian metrics associated to a contact form on a compact manifold by taking the integral of the Ricci curvature in the direction of the characteristic vector field, it is shown that on a compact regular contact manifold the only critical points of this function are the metrics for which the characteristic vector field generates a group of isometrics.

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

The study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

We characterize four-dimensional generalized complex forms and construct an Einstein and weakly *-Einstein Hermitian manifold with pointwise constant holomorphic sectional curvature which is not globally constant.

In this paper we prove a uniqueness theorem for minimal discs in R3 spanning a polygonal boundary.

We prove that a compact orientable Einstein-Thorpe manifold of dimension 8 that satisfies 6X = |P2| must be flat.

Let 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ … denote the sequence of eigenvalues of the Laplacian of a compact minimal submanifold in a unit sphere. Yang and Yau obtained an upper bound on λn+1 in terms of λn and the sum λ1 + … + λn. In this note we shall prove an improved version of this upper bound by using the method of Hile and Protter.