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general relativity - What is a manifold? - Physics Stack Exchange
Dec 11, 2016 · Here is another idea: Your manifold locally looks like a Euclidean space where the answer is simple. What if you only define your distance locally and then somehow patch it …
What is the relationship between a brane, a manifold, and a space?
Dec 25, 2015 · A manifold is a classical geometric/topological construction in mathematics, while a brane is a somewhat vague physical idea that is usually understood as being a quantum …
Why do we require manifolds to be a topological space?
Then we define to be smooth if is smooth. Here is a coordinate function. A diffeomorphism is a homeomorpism that is smooth. With this definition the manifold becomes locally diffeomorphic …
Defining a Riemannian manifold - Physics Stack Exchange
You define Rn R n by Cauchy sequences, and then you define manifolds from Rn R n. This makes many mathematicians unhappy, because they feel that something as intrinsic as a …
Why do we introduce the idea of manifold in GR books?
Jan 12, 2016 · Note, that the dynamical field in GR is not the topology of the manifold, but the pseudo-metric gνν g ν ν, which lives on the manifold. So in a sense the topological manifold is …
differential geometry - Why do we need a metric to define gradient ...
On any manifold we can define the differential df d f of a scalar f f. The differential is a 1-form: something that eats vectors and spits out scalars, or even less formally, something with one …
general relativity - Definition of spacetime in GR: what is the ...
Oct 21, 2020 · General Relativity is a mathematical framework within which we can construct Lorentzian manifold models of reality. In general, structures (e.g. the spacetime manifold) of a …
differential geometry - Does the metric define a Riemannian …
Oct 3, 2014 · In particular, topology is important. For example, just specifying that a 2-D manifold has a metric that's Euclidean everywhere is insufficient to define the manifold as being R2 R 2 …
What physically determines the point-set topology of a spacetime …
May 3, 2018 · There's no need to define the topology of the manifold from the metric. While a nice feature, the topology of the manifold is defined primarily by its atlas, which, from a physical …
Which comes first, basis vectors or coordinates?
Oct 12, 2022 · We define coordinates on the manifold, which are just homeomorphisms mapping open sets to some Rn R n, and these coordinates induce a definition of a basis on each …
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