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{{Short description|Triangle comparison theorem in Riemannian geometry}} In the [[mathematics|mathematical]] field of [[Riemannian geometry]], '''Toponogov's theorem''' (named after [[Victor Andreevich Toponogov]]) ...

{{Short description|The isometry group of a Riemannian manifold is a Lie group}} ...subsequently given by [[Richard Palais]] in 1957. The main difficulty lies in showing that a distance-preserving map, which is a priori only [[continuity ...

{{Short description|In large domains, the first Dirichlet eigenvalue of the Laplace–Beltrami opera ...ltrami operator]] is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its [ ...

...ntegral of [[Gaussian curvature]] of a non-compact [[surface (differential geometry)|surface]] to the [[Euler characteristic]]. It is akin to the [[Gauss–Bonn ...c on the manifold. Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold ''S'' with finite [[total curvature]] and finite Euler character ...

{{short description|Type of Riemannian manifold with constant Jacobi operator spectrum}} ...the volume of the intersection of two geodesic balls |journal=Differential Geometry and Its Applications}}</ref> It is named after [[Americans|American]] [[mat ...

...iguet–Puiseux theorem''' expresses the [[Gaussian curvature]] of a surface in terms of the [[circumference]] of a [[geodesic]] circle, or the area of a g ...st=Marcel | authorlink=Marcel Berger|title= A Panoramic View of Riemannian Geometry | publisher=Springer-Verlag | year=2004 | isbn=3-540-65317-1}} ...

...ian metric]] is locally determined by the [[Riemann curvature tensor]], or in other words, behavior of the curvature tensor under parallel translation de ...nnection|affine connections]] in their [[Tangent bundle|tangent bundles]], in 1959.<ref name="Library 2009 pp. 242–254">{{cite journal|last=Hicks|first=N ...

{{short description|Isometry group of a compact Riemannian manifold with negative Ricci curvature is finite}} ...Bochner]] proved in 1946 that any [[Killing vector field]] of a compact [[Riemannian manifold]] with negative [[Ricci curvature]] must be zero. Consequently the ...

In [[Riemannian geometry]], '''Schur's lemma''' is a result that says, heuristically, whenever certa Suppose <math>(M, g)</math> is a smooth [[Riemannian manifold]] with dimension <math>n.</math> Recall that this defines for each ...

...] about the topology of the space of [[Flat convergence|flat cycles]] in a Riemannian manifold. ...f null-homologous codimension 1 cycles with mod 2 coefficients on a closed Riemannian manifold Almgren isomorphism theorem implies that it is weakly homotopy equ ...

...)|discuss=Talk:Diameter of a set#Proposed merge of Diameter (computational geometry) into Diameter of a set|date=January 2025}} ...sage of diameter also occurs in medical terminology concerning a lesion or in [[geology]] concerning a rock. ...

| workplaces = [[Simons Center for Geometry and Physics]]<br />[[Stony Brook University]]<br />[[Tsinghua University]] ...a category]]. He was a permanent faculty member at the [[Simons Center for Geometry and Physics]] and a professor of mathematics at [[Stony Brook University]] ...

...ng his plenary address at the [[International Congress of Mathematicians]] in [[Nice]]. Are the equators in <math>\mathbb{S}^{n+1}</math> the only smooth embedded minimal hypersurface ...

...oriented hypersurfaces of [[Riemannian manifold]]s. In the case of curves in a two-dimensional manifold, it is identical with the [[curve shortening flo ...{mvar|n}}-dimensional manifold and let {{math|(''M'', ''g'')}} be a smooth Riemannian manifold of dimension {{math|''n'' + 1}}. Given an immersion {{mvar|f}} of ...

{{Short description|Riemannian manifold equipped with a differential p-form}} ...l]] field of [[differential geometry]], a '''calibrated manifold''' is a [[Riemannian manifold]] (''M'',''g'') of dimension ''n'' equipped with a [[differential ...

{{Short description|Class of continuous maps between Riemannian manifolds of the same dimension}} ...''R'''^''n'' of the same dimension or, more generally, between [[Riemannian manifold]]s of the same dimension, which share some of the basic properties ...

{{Short description|Problem in differential geometry}} ...Problem (Motor Control)|its possible generalization in global differential geometry|spherical Bernstein's problem}} ...

...understanding biharmonic maps was posed by [[James Eells]] and Luc Lemaire in 1983.{{sfnm|1a1=Eells|1a2=Lemaire|1y=1983|1loc=(8.7) and (8.8)}} The study Given Riemannian or pseudo-Riemannian manifolds {{math|(''M'', ''g'')}} and {{math|(''N'', ''h'')}}, a map {{mvar ...

In [[differential geometry]], the '''Minkowski problem''', named after [[Hermann Minkowski]], asks for ...re ''S''^n-1 to be the surface area measure of a [[convex body]] in <math>\mathbb{R}^n</math>. Here the surface area measure ''S_K'' ...

{{Short description|Application of differential geometry}} ...y of shape and form in [[medical imaging]]. The study of deformable shapes in CA rely on high-dimensional [[diffeomorphism group]]s <math> \operatorname{ ...