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Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...

(1992). Riemannian geometry. Basel: Birkhäuser. ISBN 978-0-8176-3490-2. Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces (Based...

Metric Structures for Riemannian and Non-Riemannian Spaces is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under...

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere...

such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold). Sub-Riemannian manifolds...

distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the...

mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described...

that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric. Because it is canonically...

Appendix C in Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes...

International Publishing, 2018. Gromov, Mikhail (1999), Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Math., vol. 152, Birkhäuser, ISBN 0-8176-3898-9...

v) > 0 for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor...

Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their standard Riemannian metrics. More examples are provided by compact...

1090/s0273-0979-1988-15654-6. Grove, Karsten (2001). "Review: Metric structures for Riemannian and non-Riemannian spaces, by M. Gromov". Bull. Amer. Math. Soc. (N.S.)...

distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C[a...

ISBN 978-3-642-23628-0. PMID 21995042. M. Gromov. Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (1999). ISBN 0-8176-3898-9 (translation...

MR 2088027. Zbl 1068.53001. Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated...

MR 2394158. Zbl 1142.53003. Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated...

is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure...

group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial...

Minkowski metric η under inclusion, is a Riemannian metric. With this metric H1(n) R is a Riemannian manifold. It is one of the model spaces of Riemannian geometry...