In 1956, he also found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold.
where is the spatial scalar curvature computed with respect to the Riemannian metric and is the extrinsic curvature,
While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to pseudo-Riemannian manifolds.
The manifolds that are used in this formulation are 4 dimensional Riemannian manifolds instead of pseudo Riemannian manifolds.
where V is four-velocity, D is the covariant derivative in the Riemannian space, and (,) is scalar product.
This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds.
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In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ : a,b → M is defined by
Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory.
The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship.
His remarkable contributions include comparison theorems of Laplacian eigenvalues on Riemannian manifolds, the maximal diameter theorem in Riemannian geometry.
where R is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold S.
Although the only Riemannian manifold with VSI property is flat space, the Lorentzian case admits nontrivial spacetimes with this property.
Riemannian manifold | manifold | Manifold | Differentiable manifold | 4-manifold | Stein manifold | Simplicial manifold | Riemann manifold | Piecewise linear manifold | Manifold Trilogy | Leek and Manifold Valley Light Railway | Finsler manifold | E8 manifold | Complex manifold | complex manifold | closed manifold | 3-manifold |
In the smooth setting, generalized flag manifolds are the compact flat model spaces for Cartan geometries of parabolic type, and are homogeneous Riemannian manifolds under any maximal compact subgroup of G.
The Weyl–Schouten theorem in mathematics says that a Riemannian manifold of dimension n with n ≥ 3 is conformally flat if and only if the Schouten tensor is a Codazzi tensor for n = 3, or the Weyl tensor vanishes for n > 3.