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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism

Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism? I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to preserve the property of being immersion. I also…
xyzzyz
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Normal curvature along a line of curvature

I have come across the following exercise (the context is curves and surfaces in $\mathbb{R}^3$ and the Gauss map): If $C=\alpha(I)$ is a line of curvature, and $k$ is its curvature at $p$, then $$ k = \mid k_n k_N \mid $$ where $k_n$ is the…
koletenbert
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pullback and pushforward examples

Where can I find some simple examples of pullbacks and pushforwards between manifolds. Specifically examples that show the details of the computations.
Wintermute
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Notation of sphere in $\Bbb{R}^n$.

In $\Bbb{R}^n$, the sphere centred at $x\in \Bbb{R}^n$ with radius $l$ is called $S^{n-1}(x,l)$ or an $(n-1)$-sphere. Why do we called it an $(n-1)$-sphere? It's certainly not defined on $\Bbb{R}^{n-1}$ instead of $\Bbb{R}^n$. Thanks in advance!
user67803
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Curvature form of Riemannian manifold in the principal bundle language

I am confused by the relation between the principal bundle and "usual" approaches to the curvature of a Riemannian manifold $M$ of dim $n$. In the "usual" approach the Riemann curvature tensor is defined as $R(X,Y)Z…
GFR
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Do differential forms need sheafification?

Let $M$ be a connected manifold. One defines differential $k$-forms as sections of the $k^{\text{th}}$ exterior power of the cotangent bundle. This is a sort of sheafification of a more naive approach, which is to let $D$ be the module of…
Michael Barz
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When is There a Solution to "Pullback Equation" of Differential Forms

All: Let $f: M \to N$ be a smooth map between manifolds, and let $w$ be a $1$-form on $M$. Under what conditions is there a $1$-form $z$ defined on $N$ so that $w=f^*z$, i.e., so that $w$ is the pullback of the form $z$ by the map $f$? All I can…
RickyBobby
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Oriented Bundle over $S^1$

Is it true that oriented bundle over $S^1$ is always trivial bundle? For example take $S^2$ and let $\gamma: S^1\to S^2$ is a great circle. As $TS^2$ is orientable, Then is it true that $\gamma^*TS^2\equiv S^1\times \mathbb R^2$.
jinu
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$\pi: E \rightarrow M$ ($E$ is a vector bundle) admits a global frame iff $E$ is a trivial vector bundle over $M$.

(Proof of backward direction is clear: if $F$ is a diffeomorphism between $E$ and $M \times \mathbb R^k$, then $p \mapsto (p, e_i) \mapsto F^{-1}(p, e_i)$ forms a basis of $E$ for $i = 1, \cdots, k$.) Proof of forward direction: Let $E$ be a vector…
James C
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Area of image under Gauss map

I am trying to solve the following question Let $\mathbf r(u,v)$ be a parametrised smooth surface in $\Bbb R^3$ with $(u,v) \in U$, a connected open subset of $\Bbb R^2$. Let $\mathbf n: U \to S^2$ be the Gauss map that assigns to each point…
Orlly
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About geodesics on a torus

Looking at some old midterms online I came across this problem and I'm having some difficulties proving it. Let $T$ be a torus of revolution paramterized by, $$x(u,v)=((r\cos u+a)\cos v,(r\cos u+a)\sin v ,r\sin u)$$ where $a,r \in \Bbb{R}$ with…
user62931
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Relationship between Gaussian, Normal and Geodesic Curvatures

How do I show that the square of the gaussian curvature is the sum of the squares of the normal and geodesic curvatures other than the one shown in page 38 of http://www.maths.lancs.ac.uk/~belton/www/notes/geom_notes.pdf? That…
anon102938
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Gauss-Bonnet theorem for spheres that almost look like a torus

[Corrected due to Jason's answer.] Imagine a torus and a flat disk fitting in the middle of its "hole" (a doughnut with a membrane in the middle). Cut the torus at its inner equator, duplicate the disk, move the two copies away from each other…
Hans-Peter Stricker
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Is the set of points in $\mathbb R^n$ with $\sum_{j=1}^n x_j^k = 0$ a submanifold?

Consider the set $$A:= \{x\in \mathbb R^n :\sum_{j=1}^n x_j^k = 0\}$$ for $k$ an odd integer. Is this a submanifold of $\mathbb R^n$ for every $n$? For $n=1$, it is just 0; for $n = 2$, it is the anti-diagonal $\{(x_1 , -x_1) : x_1 \in \mathbb R\}$,…
user15464
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Detail about the definition of orientability

I am a bit struggling with one of the many definitions of orientability. In what follows $M$ will always denote a smooth, connected manifold, $T_{m}M$ will be the tangent space at $m$, $\mathcal{B}_{m}$ will be the set of ordered bases of $T_{m}M$,…
MWL
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