Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.
The first author and F. Prufer gave an explicit classification of all Riemannian 3-manifolds with distinct constant Ricci eigenvalues and satisfying additional geometrical conditions. The aim of the present paper is to get the same classification under weaker geometrical conditions.
[For the entire collection see Zbl 0699.00032.] A connection structure (M,H) and a path structure (M,S) on the manifold M are called compatible, if locally where and express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, whence it follows not that H is linear, i.e. if a connection structure is compatible with a spray, then...
Let Mⁿ (n ≥ 3) be an n-dimensional complete hypersurface in a real space form N(c) (c ≥ 0). We prove that if the sectional curvature of M satisfies the following pinching condition: , where δ = 1/5 for n ≥ 4 and δ = 1/4 for n = 3, then there are no stable currents (or stable varifolds) in M. This is a positive answer to the well-known conjecture of Lawson and Simons.
We obtain conditions under which a submanifold of a Poisson manifold has an induced Poisson structure, which encompass both the Poisson submanifolds of A. Weinstein [21] and the Poisson structures on the phase space of a mechanical system with kinematic constraints of Van der Schaft and Maschke [20]. Generalizations of these results for submanifolds of a Jacobi manifold are briefly sketched.
Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces and classified all of them with a constant radius of the indicatrix. We discuss the mentioned results from the point of view of the twistor theory, providing some new proofs. It turns out that the superminimal surfaces investigated by geometers at the beginning of this century as well as by O. Boruvka...