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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).

32835 questions
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A question about Gauss Bonnet theorem

If $S$ is a surface which is the complement of finitely many points in a compact surface, and the metric in $S$ is complete, then is Gauss-Bonnet theorem still valid for $S$?
hao
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1 answer

Reparametrize the curve problem in Differential geometry

Reparametrize the curve $\alpha(t)=(e^{t},e^{-t},\sqrt{2}t), \; \alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$, using $h(s)= \log(s)$ on $J:s>0$. Check the equation in Lemma in this case by calculating each side separately. This Lemma : If $\beta$…
Esteban
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Proving the contracted Bianchi identity?

From Lee's book, the differential Bianchi identity states that for the Riemann curvature tensor, $$R_{ijkl;m} + R_{ijlm;k} + R_{ijmk;l}=0$$ The the proof is, contract on $i, l$, and then on $j, k$ after raising an index on each pair. Then we obtain…
user9812334
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The proof of cartan's magic formula

I want to use integral to prove the cartan's magic formula, i.e., it's enough to prove that for all small disk $D$ of dim=k=deg $\alpha$ in a manifold $M$, we have $$ \int_D L_X\alpha = \int_D (d(i_X\alpha)+i_X d\alpha. $$ To prove that, we…
Yui
  • 401
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Extension of mean-curvature normal

Suppose $M$ is a two-dimensional manifold with metric $\bar{g}$, and $r: M \to \mathbb{R}^3$ is a (not necessarily isometric) embedding of $M$ into $\mathbb{R}^3$ with first fundamental form $g$ and second fundamental form $b$. I know…
user7530
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Tangent line to a differentiable curve

Let $\alpha: I \to \mathbb{R}^{n}$ be a differentiable curve such that $\alpha'(a) \neq 0$ for some $a \in I$. The line $L \subset \mathbb{R}^{n}$ through $\alpha(a)$ is the tangent line to the curve $\alpha$ at that point if and only if $$ \lim_{t…
limber
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Derivative of a vector valued function under an orthogonal transformation

Let $\alpha : \mathbb{R} \to \mathbb{R}^3$ be a space curve. I'm trying to show that its curvature, torsion, and arc length are invariant under orthogonal transformations. If $\rho: \mathbb{R}^3\to \mathbb{R}^3$ is an orthogonal transformation,…
Sergei Bobrovsky
  • 63
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Can we define a induced metric like this?

Let $\Sigma_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric $g$, $\{\frac{\partial}{\partial x^i}\}, 1\leq i\leq3$ is the standard coordinate frame in $\mathbb{R}^3$, If the unit normal…
Jer
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1 answer

Is a surjective submersion map proper?

Let $M$ and $N$ be two smooth manifolds and $f$ a surjective smooth map from $M$ to $N$ which is a submersion. If for any $p \in N$, $f^{-1}(p)$ is compact, then is $f$ necessarily proper, that is, the preimage of compact set is compact?
Summer
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Orientability of Möbius band

Can we cover the Möbius band with a finite atlas such that the determinant of the Jacobian of each transition map is negative everyhwere?
Joel J.
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Compatible connection on the associated vector bundle

Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says "holomorphic") structure of $E$. Let $P\rightarrow…
Flavius Aetius
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The differential of any chart on a manifold $M$ is an isomorphism of $M$'s tangent space

My book asserts that a chart $(U,h)$ on an $n$-dimensional manifold $M$ induces an isomorphism of it's tangent space at a point $p$ and the tangent space of $\mathbb R^n$ at the image $h(p)$ via it's differential $T_ph:T_pU\to T_{h(p)}h[U]$. How is…
superAnnoyingUser
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Formula for the torsion of a regular curve parametrized by arc length

Let $\alpha:\, I \to \mathbb{R}^3$ be a curve parametrized by arc length $s$, with curvature $k(s) \neq 0$, for all $s \in I$. Show that the torsion $\tau$ of $\alpha$ is given by: $$ \tau(s) = -\frac{\alpha'(s) \times \alpha''(s) \cdot…
Sergei Bobrovsky
  • 63
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1 answer

Proof of area or volume element

Why is the volume element equal to the square root of the absolute value of the metric tensor determinant, i.e. $$dV=\sqrt{\left | g \right |} dx^0dx^1dx^2$$
BinaryBurst
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Osculating plane<=>distance limit

Let $r$ be a unit-speed bi-regular curve. (It passes the point $s_0$) Let $distP(q)$ be the distance between the plane $P$ and the point $q$. Question. The plane is equal to the osculating plane of $r$ at $s_0$ if and only if $P$ contains…
Math-Nerd
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