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The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry , an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport , covariant derivatives ...
The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field , the matrix () is skew-symmetric; equivalently, it is an element of the Lie algebra (). This can be seen as follows.
the connection is torsion-free, i.e., T ∇ is zero, so that ∇ X Y − ∇ Y X = [X, Y]; parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved. This connection is called the Levi-Civita connection. The term "symmetric" is often used instead of torsion-free for the first property.
The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the e α , it is the unique k × k matrix ( ω α β ) of one-forms on M such that
A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. [17] This concept has become one of the most powerful tools of theoretical physics , as it has become evident that practically all laws of nature originate in symmetries.
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.