We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in th...

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, oriented, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle. Moreover let ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m com...

In this paper we prove a new geometric integrability theorem for almost complex manifolds. Let M be a connected, smooth 2m real dimensional manifold with an U(m) structure on its tangent bundle and ∇ be a compatible connection on it. Assume the complexified curvature of the connection has vanishing (0, 2) component. Then we claim that M can be given the structure of an m complex dimensional Käh...

We develop the Palatini formalism within framework of generalized Riemannian geometry Courant algebroids. In this context, variation a Einstein–Hilbert–Palatini action - formed using metric, algebroid connection (in contrary to ordinary case, not necessarily torsionless one) and volume form leads naturally proper notion Levi-Civita low-energy effective actions string theory.

In this paper, we study locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of metric. As main result, classify such being not flat particular, $2, 3$-dimensional are completely determined.

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFTtype correspondence between classical 2 + 1 dimensional vacuum general relativity theory on Σ× R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ, a compact, oriented Riemann surface of genus greater than one. More precisely, we prove that if over Σ with a fi...

A noncommutative-geometric generalization of the classical formalism of frame bundles is developed, incorporating into the theory of quantum principal bundles the concept of the Levi-Civita connection. The construction of a natural differential calculus on quantum principal frame bundles is presented, including the construction of the associated differential calculus on the structure group. Gen...

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with ...

Since its introduction by Élie Cartan, the holonomy of a connection has played an important role in differential geometry. One of the best known results concerning holonomy is Berger’s classification of the possible holonomies of Levi-Civita connections of Riemannian metrics. Since the appearance of Berger [1955], much work has been done to refine his list of possible Riemannian holonomies. See...

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A...