# Tagged Questions

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### What does “flat hypersurface” mean?

If $S$ is a flat hypersurface with boundary in $\mathbb{R}^n$, what does it mean? Is it just a simple open domain (found in most PDE contexts)?
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### Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...
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### Existence of differential form on a manifold

I have a fundamental question about the existence of differential forms on manifolds. A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a ...
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### Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
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### Is $S^0$ a manifold?

Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does ...
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### Example of a manifold?

Why is this picture an example of a $1$-dimensional manifold? My thought process is: the circle must have a point removed from it because otherwise it would be self-intersecting, and self-intersection ...
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### geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
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### Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.
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### Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$

In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n} X^{i} \frac{\partial}{\partial x^{i}}, X^{i} \in$$C^{\infty}(p)$. Then 1.The differentiation of ...
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### What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
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### Defintion of totally geodesic flat submanifold

I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian ...
Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
Consider the following definition: Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...