Levi-Civita connection

Calculus on Manifolds

Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018

Based on this embedding, compute the following: The components gab of the induced metric on the torus resulting from the standard Euclidean metric in ℝ3.The Christoffel symbols of the Levi-Civita connection.The independent non-zero component Rθφθφ of the curvature tensor.The components of the Ricci tensor Rab and the Ricci scalar R. Verify that R is not constant and that it has a different sign on different points in the manifold.

Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds

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Journal Information

Published in Applicable Analysis, 2022

Jingjing Fan, Xiaolong Qin, Bing Tan

Let ∇ be the Levi–Civita connection associated with the Riemannian metric. Let γ be a smooth curve in . A vector field X is said to be parallel along γ iff . If is parallel along γ, i.e. , then γ is said to be geodesic, and in this case, is a constant. Furthermore, if , then γ is called normalized. A geodesic joining x to y in is said to be minimal if its length equals . Let be a geodesic and denote the parallel transport along γ with respect to V, which is defined by for all and , where V is the unique vector field satisfying and . Then, for any , is an isometry from to . We will write instead of in the case that γ is a minimal geodesic joining x to y if this will avoid any confusion.

Levi-Civita connection

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Calculus on Manifolds

Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds