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In differential geometry, a -structure is an important type of G-structure that can be defined on a smooth manifold. If M is a smooth manifold of dimension seven, then a G 2 -structure is a reduction of structure group of the frame bundle of M to the compact , exceptional Lie group G 2 .
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G -structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields .
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of R n {\displaystyle {\mathbb {R} }^{n}} , with monodromy acting by affine transformations .
The study of calculus on differentiable manifolds is known as differential geometry. "Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth (which itself has many meanings), and analytic.
An example of a Riemannian submersion arises when a Lie group acts isometrically, freely and properly on a Riemannian manifold (,). The projection π : M → N {\displaystyle \pi :M\rightarrow N} to the quotient space N = M / G {\displaystyle N=M/G} equipped with the quotient metric is a Riemannian submersion.
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor.
Spivak, Michael (1999) A Comprehensive Introduction to Differential Geometry (3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels.
Let (,,,) be an almost-contact manifold.One says that a Riemannian metric on is adapted to the almost-contact structure (,,) if: = =. That is to say that, relative to , the vector has length one and is orthogonal to (); furthermore the restriction of to is a Hermitian metric relative to the almost-complex structure | ().