Here the matrix (gij) (assumed nondegenerate) defines a pseudo-Riemannian metric (with upper indices) of zero curvature on the u-space, Fjk i = Fjki(u) being the corresponding Levi-Civita connection. Thus, the integrability condition can be formulated in terms of the differential geometry of SHT. For such integrable systems S. P. Tsarev [3] found a generalization (for N _> 3) of the hodograph m...

In this paper the Berry and Aharonov-Bohm phases are generalized to nonlinear topological phase fields on pseudospheres, where the coordinate vector field is parallel transported along the signal/soliton vector field with Levi–Civita connection. Projective PSL(2,R) symmetry describes the relativistic self-interacting bosonic sine-Gordon field. A Coulomb potential can be induced as the stereogra...

Abstract. This paper presents a mathematical view of aspects of physics, showing how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. This work is motivated by discrete calculus, as it is shown that classical discrete calculus embeds in a non-commutative context. It is shown how various processe...

In this paper, we continue the analysis of heterotic string compactifications on half-flat mirror manifolds by including the 10-dimensional gauge fields. It is argued, that the heterotic Bianchi identity is solved by a variant of the standard embedding. Then, the resulting gauge group in four dimensions is still E6 despite the fact that the Levi-Civita connection has SO(6) holonomy. We derive t...

1. Integration on manifolds. 1 2. The extension of the Levi-Civita connection. 4 3. Covariant derivatives. 7 4. The Laplace operator. 9 5. Self-adjoint extension of the Laplace operator. 11 6. The Poincaré Lemma. 13 7. de Rham Theorem. 14 8. Formal Hodge Theorem. 16 9. The Hodge theorem. 17 10. Proof of the Hodge Theorem. 17 11. More about elliptic regularity. 22 12. Semigroups and their genera...

The Levi-Civita connection and geodesic equations for a stationary spacetime are studied in depth. General formulae which generalize those for warped products are obtained. These results are applicated to some regions of Kerr spacetime previously studied by using variational methods. We show that they are neither space-convex nor geodesically connected. Moreover, the whole stationary part of Ke...

ABSTRACT. We present a short-time existence theorem of solutions to the initial value problem for Schrödinger maps of a closed Riemannian manifold to a compact almost Hermitian manifold. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a lo...

We present a compared analysis of some properties of indefinite almost S-manifolds and indefinite S-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and φ-sectional curvature of indefinite almost S-manifolds and state an expression of the curvature tensor field for the indefinite S-space forms. We an...

We consider invariant symplectic connections ∇ on homogeneous symplectic manifolds (M, ω) with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If M is compact with finite fundamental group then (M...

Abstract A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian of ambient space. This allows us to define, for any nonzero number k , so-called -th generalized Tanaka–Webster connection. With this connection and Levi-Civita one we can associate two tensors type (1,2) Jacobi operator $$R_{\xi }$$ <mml:math xmlns:mml="http://www.w3.org/...