Can every Riemannian manifold be embedded in a sphere?
2016年5月19日 · The maps you are constructing will not be homeomorphisms to their images (in the sphere) if the Riemannian manifold is noncompact (at least in general). What you getting are isometric injective immersions. One needs to work a bit more to get embeddings.
real analysis - Illustrative Example of Riemannian Manifold ...
2021年1月2日 · Tour Start here for a quick overview of the site
metric spaces - Completion of a riemannian manifold.
2017年11月22日 · If we remove the cone point, what's left can be given the structure of a smooth Riemannian manifold in such a way that the inclusion map is an isometric embedding in the metric sense. Then the cone on $\mathbb{R}P^2$ is the completion, so is not a manifold. $\endgroup$ –
general topology - Topologies in a Riemannian Manifold
The proof boils down to showing that the Riemannian metric is uniformly comparable to the Euclidean metric in small coordinate balls. You can read a proof in my Introduction to Smooth Manifolds (2nd ed., Theorem 13.29).
riemannian geometry - Constant curvature riemann manifold
By contrast, there are moduli spaces of Einstein metrics even on a single smooth manifold. For example, the smooth $4$-manifold underlying a complex quartic in complex projective $3$-space admits, up to scaling, a $20$-dimensional family of Ricci-flat Kähler metrics by Yau's solution of the Calabi conjecture.
Riemannian measure and Hausdorff measure in a general …
Let $ M $ be a Riemannian manifold and let $ \\mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact supp...
Differential and Riemannian structure on the cone
2022年4月15日 · This can, in particular, be applied to manifolds. What one gets is a topological space which, in general, is not a manifold, but can, in some special cases, be given a structure of a smooth manifold (see for example the discussion here).
Does every manifold M always admit a Riemannian metric?
2015年9月8日 · A manifold may admit more than one Riemannian metric. It becomes a Riemannian manifold when we pick one. Otherwise we don't have a particular Riemannian metric in mind, so it's just a manifold. $\endgroup$ –
How does one complexify a real $n$-dimensional Riemannian …
2018年7月24日 · Idea: The notion of complexification for Lie groups exists, so perhaps one can "complexify" a real Riemannian manifold by realizing it as a Lie group (or the quotient of one). It seems that under complexification of a real manifold some additional information must be added to determine a complex structure.
The canonical measure on a Riemannian manifold
2022年2月9日 · It seems well-known that the volume form natually induces a canonical measure on a Riemannian manifold. For example, this page mentions this fact. I am wondering the construction of such a measure. For my background, I learned some real analysis and read Tu's An Introduction to Manifolds.