Foundations of Differential Geometry, Volume I.

By F. Linton, Shôshichi Kobayashi, K. Nomizu. 1965

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Citations in the corpus

Gauge Equivariant Convolutional Networks and the Icosahedral CNN (2019, neutral): “For further study, we recommend (Sharpe, 1997; Shoshichi Kobayashi, 1963; Husem¨oller, 1994; Steenrod, 1951; Wendl, 2008; Marsh, 2016; Crane, 2014).”
Normalizing Flows for Probabilistic Modeling and Inference (2019, background): “The map T induces a metric G(u) on the tangent space of X at x = T (u), given by (Kobayashi and Nomizu, 1963): G(u) = JT (u) JT (u).”
SE-Sync: A certifiably correct algorithm for synchronization over the special Euclidean group (2016, methodology): “We first exploit this problem’s low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold [12, 48], and then design a fast Riemannian optimization method to solve this reduction efficiently.”
Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds (2018, background): “In fact, the exponential map deﬁnes a one-to-one correspondence between T x,δ M and M x,δ if δ is smaller than the injective radius of M [18, 7].”
Optimal and Cooperative Control of Vehicle Formations (2002, background): “See [35, 58] for extensive development of these concepts and their relation to Riemannian geometry.”
A Novel Space-Time Representation on the Positive Semidefinite Cone for Facial Expression Recognition (2017, background): “Bonnabel and Sepulchre deﬁne a connection (see [22, p. 63]) on the principal bundle Π : V n, 2 × P 2 → S + (2 , n ) by setting the horizontal subspace H ( U,R 2 ) at the point ( U, R 2 ) to be the space of tangent vectors ( M, N ) such that M T U = 0 and N is an arbitrary 2 × 2 symmetric matrix.”
Local BRST cohomology in gauge theories (2000, background): “[119, 159, 226]): (i) Every G-invariant polynomial in the F I A is a polynomial P (f1(FA), .”
On the present state of the Andersen-Lempert theory (2010, background): “, see [48]) the Lie algebra generated by completely integrable algebraic vector fields contains only fields of divergence zero with respect to ω.”