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Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor .
Download as PDF; Printable version; ... Pages in category "Riemannian manifolds" The following 41 pages are in this category, out of 41 total. ... Simons' formula ...
M, g) denotes a pseudo-Riemannian manifold. TM is the tangent bundle of M. g is the pseudo-Riemannian metric of M. X, Y, Z are smooth vector fields on M, i. e. smooth sections of TM. [X, Y] is the Lie bracket of X and Y. It is again a smooth vector field. The metric g can take up to two vectors or vector fields X, Y as arguments.
In the case of a Riemannian 2-manifold, the fundamental theorem of Riemannian geometry can be rephrased in terms of Cartan's canonical 1-forms: Theorem. On an oriented Riemannian 2-manifold M, there is a unique connection ω on the frame bundle satisfying
In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the Jacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensional Riemannian manifold.