We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of (J2 = ±1)-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among con...

If ∇ is a torsionless connection on the tangent bundle of a manifold M the Weyl curvature W is the part of the curvature in kernel of the Ricci contraction. We give a coordinate free proof of Weyl’s result that W vanishes if and only if (M,∇) is (locally) diffeomorphic to RP with ∇, when transported to RP, in the projective class of ∇LC , the Levi-Civita connection of the Fubini–Study metric on...

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.

We investigate bi–Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non–stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric–induced nonlinear connection (N–connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a ...

On an almost Hermitian manifold, there are two scalar curvatures associated with a canonical connection. In this paper, explicit formulas on these obtained in terms of the Riemannian curvature, norms components covariant derivative fundamental 2-form respect to Levi-Civita connection, and codifferential Lee form. Then we use them get characterization results Kähler metric, balanced locally conf...

1. Smooth Manifolds: 9/22/14 1 2. Going on a Tangent (Space): 9/24/14 1 3. Immersions and Embeddings: 9/26/14 3 4. Riemannian Metrics: 9/29/14 3 5. Indiana Jones and the Isometries of the Upper Half-Plane: 10/1/14 5 6. That's an Affine Connection You've Got There: 10/3/14 7 7. Missed (Levi-Civita) Connections: 10/6/14 9 8. The Levi-Civita Connection and the Induced Metric: 10/8/14 9 9. Geodesic...

The coadjoint orbits of compact Lie groups carry many Kähler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi–Civita connection of the Riemannian metric, and we use the complex structure to give a fairly explicit construction of the Dirac operator for the Riemannian metric, in a way that avoids use of the spin groups. Subst...

We define a Hesse soliton, that is, self-similar solution to the flow on Hessian manifolds. On information geometry, e-connection is important, which does not coincide with Levi–Civita one. Therefore, it interesting consider manifold flat connection call proper manifold. In this paper, we show any compact soliton expanding and non-trivial gradient proper. Furthermore, dual space of Hesse–Einste...

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coo...