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is also a Riemannian metric on . We say that ~ is (pointwise) conformal to . Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric.
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque [1] [2].Torsion could be defined as strain [3] [4] or angular deformation [5], and is measured by the angle a chosen section is rotated from its equilibrium position [6].
Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness.
This proves the uniqueness of a torsion-free and metric-compatible condition, since if g(W, Z) is equal to g(U, Z) for arbitrary Z, then W must equal U. This is a consequence of the non-degeneracy of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence of g kl.
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. [1] This is equivalent to: A connection for which the covariant derivatives of the metric on E ...
The torsion form, an alternative characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω , a gl ( n )-valued one-form which maps vertical vectors to the generators of the right action in gl ( n ) and equivariantly intertwines the right action of GL( n ) on the ...
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic ...