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Surfaces with Riemannian Metric
2021The best we can do is to set up a list of axioms, say wishes, that makes \( \nabla_{\textit{X}} \textit{Y} \) look like a directional derivative. An optimal list of wishes would be such that one can fulfill these wishes but only in a unique way.
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The Notion of a Riemannian Metric
1997In the preceding chapter we associated with each curvilinear coordinate system x = (x 1, ... , x n) on a domain C a smooth functional matrix G(x), and these matrices, at each point, behave as a quadratic form under a change-of-coordinate transformations.
Anatolij T. Fomenko, Tosiyasu L. Kinii
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2017
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the "shape" of a manifold.
Bustamante, Mauricio +1 more
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Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the "shape" of a manifold.
Bustamante, Mauricio +1 more
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The Riemannian Space of Kähler Metrics
2011The goal of this lecture is to describe the Riemannian structure on the space of Kahler metrics (in a fixed cohomology class) on a compact complex manifold which was proposed by Mabuchi [Mab87] and further stud- ied by Semmes [Sem92], Donaldson [Don99], Chen and Calabi [Che00,CC02].
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Journal of Mathematical Sciences, 2007
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The Quarterly Journal of Mathematics, 2000
The extrinsic complexification \({\mathcal M}\) of a real analytic Riemannian manifold \((M,g)\) provides a Grauert tube equipped with an anti-holomorphic involution \(\sigma\) fixing \(M\) together with a strictly plurisubharmonic function \(\rho\) satisfying the complex Monge-Ampère equation \((\partial\overline{\partial})^n\rho=0\) in \({\mathcal M}\
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The extrinsic complexification \({\mathcal M}\) of a real analytic Riemannian manifold \((M,g)\) provides a Grauert tube equipped with an anti-holomorphic involution \(\sigma\) fixing \(M\) together with a strictly plurisubharmonic function \(\rho\) satisfying the complex Monge-Ampère equation \((\partial\overline{\partial})^n\rho=0\) in \({\mathcal M}\
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On Indefinite Binary Riemannian Metrics
American Journal of Mathematics, 1955openaire +1 more source