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What is a Manifold? - Mathematics Stack Exchange
WEBMar 20, 2015 · Formally speaking, a (differentiable) d-dimensional manifold X is a topological space where each point x has a neighborhood that is topologically equivalent (homeomorphic) to a d-dimensional Euclidean space, called the tangent space. Share. Cite. answered Feb 9, 2021 at 10:44. Gonçalo Peres.
Math.stackexchange.comWhat exactly is a manifold? - Mathematics Stack Exchange
WEB2. A manifold is some set of points such that for each one we can consult a chart which will transport some region of that manifold containing the point into a region of euclidean space (well understood). A country is a region of the Earth's surface. A map of a country is a chart that gives you that region of the manifold (Earth) projected onto
Math.stackexchange.comIntroductory texts on manifolds - Mathematics Stack Exchange
WEBJun 29, 2022 · However, I would argue that one of the best introductions to manifolds is the old soviet book published by MIR, Mishchenko/Fomenko - "A Course of Differential Geometry and Topology". It develops everything up from Rn R n, curves and surfaces to arrive at smooth manifolds and LOTS of examples (Lie groups, classification of surfaces, …
Math.stackexchange.comDefinition of closed, compact manifold and topological spaces
WEBJun 15, 2019 · A "closed manifold" is a topological space that has the following properties: it is a manifold [locally Euclidean, second countable, Hausdorff topological space] that is additionally compact and without boundary. However, this is distinct from a "closed set" in topology, which can change depending on the embedding. $\endgroup$ –
Math.stackexchange.comWhat is the difference between a variety and a manifold?
WEBNov 5, 2010 · The classical notion of a manifold is simply a locally closed subset of $\mathbf{P}^n(\mathbf{R})$ (or just $\mathbf{R}^n$, although looking at locally-closed subsets of projective space makes the analogy that much stronger), and the notion of a variety (quasi-projective, I guess) comes from considering locally closed subsets of a …
Math.stackexchange.comDefinition of manifold - Mathematics Stack Exchange
WEBThe broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. A topological manifold is a topological space locally homeomorphic to a Euclidean space. In both concepts, a topological space is homeomorphic to another topological space with richer structure than just topology
Math.stackexchange.comWhat is an example of a manifold that is not smooth?
WEBAug 4, 2021 · Consider a triangle, which is a topological manifold that is homeomorphic to the circle but not differeomorphic as it has corners. And so is a pentagon. For the K10 example, if I remove a point it would be smoothable. Kervaire's example admits the following relation: H ∗ (K10) = H ∗ (S5 × S2)
Math.stackexchange.comProving that the general linear group is a differentiable manifold
WEBWe know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold.
Math.stackexchange.comIs every manifold a metric space? - Mathematics Stack Exchange
WEBNov 15, 2015 · The answer is "it depends on your definition". If your definition of "manifold" doesn't stipulate second countable (the topology is generated by countably many open sets), there exist manifolds such as the long line whose topology is not metrizable. Similarly, a non- Hausdorff manifold is not metrizable. If your definition stipulates (M, T) (M
Math.stackexchange.comInfinite-dimensional manifolds: Fréchet, Banach and Hilbert …
WEBJul 15, 2022 · 2. There are different generalizations of the concept of differentiable manifold (usually of finite dimension) to the case with infinite dimension. Essentially, it is sought that these manifolds with infinite dimension are homeomorphic, as these topological spaces, to vector spaces of infinite dimension, and this gives rise to the following
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