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Results tagged with riemannian-geometry
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user 90512
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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Non-negatively curved manifolds and the volume of balls
Whether a complete non-compact non-flat Riemannian $n$-manifold $M$ with non-negative sectional curvature has Euclidean volume growth?
That is, whether there is a constant $C>0$ such that $\mathrm{Vo …
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11
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Can the diameter be controled by the injectivity radius and the volume?
Diameter bounded from above is usually needed in the finiteness theorem or other convergence theorems in Riemannian Geometry. Let $M^n$ be a closed manifold and {$g_i$} be a family of smooth Riemannia …
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Harmonic map in the homotopy class of the identity map
Eells and Sampson's existence Theorem states that if $(N, h)$ is nonpositively curved, then a given map $f : (M, h') \to (N, h)$ can be deformed into a harmonic map in its homotopy class. Here smooth …
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Non-positive sectional curvature in 3-dimensional manifold
The answer of the following question may be well-known in the field of Geometric Topology, so I ask for help in here.
Does the total space of circle bundle over a closed hyperbolic surface admit a …
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11
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1
answer
440
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Scalar curvature and the degree of symmetry
Let $M$ be a closed connected smooth manifold. We define the degree of symmetry of $M$ by $N(M):=\sup_\limits{g}\mathrm{dim}\,\mathrm{Isom}(M,g)$, where $g$ is a smooth Riemannian metric on $M$ and $ …
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3
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If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non...
If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature?
I wish to use the result about the question and find Leeb's work 3-manifol …
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3
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What can be reflected by the $C^0$-limit of Riemannian metrics?
Let $M^n$ be a closed connected smooth manifold and {$g_i$} be a family of smooth Riemannian metrics on it such that $g_i$ $C^0$-converges to the smooth Riemannian metric $g$ on $M^n$.
Can it possi …
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Existence of harmonic maps onto the $n$-sphere
Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g_{st})$ be the standard round $n$-sphere.
Whether there exists a non- …
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