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...

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with deformed Sasaki metric. In this paper, firstly we investigate all forms of curvature tensors (Riemannian tensor, Ricci curvature, sectional scalar curvature). Secondly, study the geometry unit metric, where presented formulas Levi-Civita connection also

Let (M, g) be an n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G
 and called Mus-Cheeger-Gromoll metric on tangent bundle TM. We calculate its Levi-Civita connection curvature tensor. Also, study geometry (TM, G).

We develop a theory in which there are couplings amongst Dirac spinor, dilaton and non-Riemannian gravity and explore the nature of connection-induced dilaton couplings to gravity and Dirac spinor when the theory is reformulated in terms of the Levi-Civita connection. After presenting some exact solutions without spinors, we investigate the minimal spinor couplings to the model and in conclusio...

We study the quantum sphere Cq [S] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω ⊕ Ω in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci ...

In the present paper, we discuss singular minimal surfaces in Euclidean $3-$space $\mathbb{R}^{3}$ which are minimal. Such a surface is nothing but plane, trivial outcome. However, non-trivial outcome obtained when modify usual condition of minimality by using special semi-symmetric metric connection instead Levi-Civita on $\mathbb{R}^{3}$. With this new connection, prove that, besides planes, ...

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...

Preliminary results concerning non-quadratic (and non-bijective) transformations that exibit a degree of parentage with the well known Levi-Civita, Kustaanheimo-Stiefel, and Fock transformations are reported in this article. Some of the new transformations are applied to non-relativistic quantum dynamical systems in two dimensions.