Orthonormal frame
Encyclopedia
In Riemannian geometry
and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold
equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is an ordered basis of the tangent space
at P consisting of vectors which are orthonormal with respect to the bilinear form gP.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is an ordered basis of the tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
at P consisting of vectors which are orthonormal with respect to the bilinear form gP.
See also
- Frame (linear algebra)Frame (linear algebra)In mathematics, a frame of a vector space V, is either of two distinct notions, both generalizing the notion of a basis:* In one definition, a k-frame is an ordered set of k linearly independent vectors in a space; thus k ≤ n the dimension of the vector space, and if k = n an...
- Frame bundleFrame bundleIn mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
- k-frameK-frameIn linear algebra, a branch of mathematics, a k-frame is an ordered set of k linearly independent vectors in a space; thus k ≤ n, where n is the dimension of the vector space, and if k = n an n-frame is precisely an ordered basis.If the vectors are orthogonal, or orthonormal,...
- Moving frameMoving frameIn mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.-Introduction:...