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Riemannian metrics constructed this way are left-invariant; right-invariant Riemannian metrics could be constructed likewise using the right multiplication map instead. The Levi-Civita connection and curvature of a general left-invariant Riemannian metric can be computed explicitly in terms of g e , the adjoint representation of G , and the Lie ...
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x 1, ..., x n) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.
The Calabi flow was introduced by Eugenio Calabi in 1982 as a suggestion for the construction of extremal Kähler metrics, which were also introduced in the same paper. It is the gradient flow of the Calabi functional; extremal Kähler metrics are the critical points of the Calabi functional.
The r* cross-correlation metric is based on the variance metrics of SSIM. It's defined as r*(x, y) = σ xy / σ x σ y when σ x σ y ≠ 0, 1 when both standard deviations are zero, and 0 when only one is zero. It has found use in analyzing human response to contrast-detail phantoms. [18] SSIM has also been used on the gradient of ...
An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.
In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold.First introduced by Richard S. Hamilton, [1] Yamabe flow is for noncompact manifolds, and is the negative L 2-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric ...
On functions, it agrees with the operator given as the divergence of the gradient. If the connection of interest is the Levi-Civita connection one can find a convenient formula for the Laplacian of a scalar function in terms of partial derivatives with respect to a coordinate system: