# Tagged Questions

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### $f:M\to N$ smooth manifold map. $F(x)=(x,f(x))\in M\times N$. For each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$.

Let $M$ and $N$ be to manifolds and $f:M\to N$ smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$. ...
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### About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
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### Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
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### “Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
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### Is this map smooth?

Let $M$ and $N$ be smooth manifolds and $$f:M\times N\to \mathbb{R}$$ a map. Suppose that the maps $$M\to\mathbb{R},\quad p\mapsto f(p,q_0)$$ $$N\to\mathbb{R},\quad q\mapsto f(p_0,q)$$ are smooth for ...
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### Relation between nonorientability of the Möbius strip and the Möbius bundle

There are two ways in which the open Möbius strip $M$ is related to orientability: $M$ is nonorientable as a manifold; $M$ is the total space of the nonorientable line bundle $M \to S^1$. Is there ...
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### Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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### Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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### Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
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### How to prove that a space is not a differential manifold?

Given a box （the surface of a cubic） in R^3 space, can I give a smooth structure on it to make it a differential manifold?