scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

CAN TORSION BE TREATED AS JUST ANOTHER TENSOR FIELD?

2012 ◽  
Vol 07 ◽  
pp. 158-164 ◽  
Author(s):  
JAMES M. NESTER ◽  
CHIH-HUNG WANG
Keyword(s):  
Gauge Theory ◽  
Tensor Field ◽  
Affine Connection ◽  
Cartan Geometry ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Alternative Gravity Theories ◽  
Theory Of Gravity ◽  
Gauge Theory Of Gravity ◽  
Alternative Gravity

Many alternative gravity theories use an independent connection which leads to torsion in addition to curvature. Some have argued that there is no physical need to use such connections, that one can always use the Levi-Civita connection and just treat torsion as another tensor field. We explore this issue here in the context of the Poincaré Gauge theory of gravity, which is usually formulated in terms of an affine connection for a Riemann-Cartan geometry (torsion and curvature). We compare the equations obtained by taking as the independent dynamical variables: (i) the orthonormal coframe and the connection and (ii) the orthonormal coframe and the torsion (contortion), and we also consider the coupling to a source. From this analysis we conclude that, at least for this class of theories, torsion should not be considered as just another tensor field.

The affine connection

2021 ◽  
pp. 121-132
Author(s):  
Andrew M. Steane
Keyword(s):  
Covariant Derivative ◽  
Affine Connection ◽  
Connection Coefficients ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Coordinate Basis ◽  
Basis Vectors

The connection and the covariant derivative are treated. Connection coefficients are introduced in their role of expressing the change in the coordinate basis vectors between neighbouring points. The covariant derivative of a vector is then defined. Next we relate the connection to the metric, and obtain the Levi-Civita connection. The logic concerning what is defined and what is derived is explained carefuly. The notion of a derivative along a curve is defined. The emphasis through is on clarity and avoiding confusions arising from the plethora of concepts and symbols.

The Levi-Civita Connection

2022 ◽  
pp. 117-167
Author(s):  
Joel W. Robbin ◽  
Dietmar A. Salamon
Keyword(s):  
Civita Connection ◽  
Levi Civita Connection

scholarly journals Covariant derivatives on Kähler C-spaces

1996 ◽  
Vol 143 ◽  
pp. 31-57
Author(s):  
Koji Tojo
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Levi Civita Connection

Let (M, g) be a Kähler C-space. R and ∇ denote the curvature tensor and the Levi-Civita connection of (M, g), respectively.In [6], Takagi have proved that there exists an integer n such that

scholarly journals On the Koszul formula in noncommutative geometry

2020 ◽  
Vol 32 (10) ◽  
pp. 2050032 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Scalar Curvature ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Civita Connection ◽  
Canonical Metric ◽  
Levi Civita Connection

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

2D RICCI FLAT GRADIENT SOLITONS ARISING FROM REMARKABLE MODELS IN PHYSICS

2013 ◽  
Vol 10 (10) ◽  
pp. 1350059
Author(s):  
GABRIEL BERCU
Keyword(s):  
Riemannian Manifold ◽  
Local Computation ◽  
Ricci Flat ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Gradient Solitons

On a pseudo-Riemannian manifold (M, g) we consider ∇ the Levi-Cività connection associated to metric g and a function f : M → ℝ whose pseudo-Riemannian Hessian [Formula: see text] is non-degenerate and with constant signature. We study properties of the pseudo-Riemannian manifold (M, h), [Formula: see text] in terms of local computation. Investigating the conditions for existence of the equal connections [Formula: see text], produced by g and h, we determine classes of explicit Ricci flat gradient solitons for some particular forms of g, arising from remarkable physics models.

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