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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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Spivak problem on orientations. (A comprehensive introduction to differential geometry)

I have a problems doing exercise 16 of chapter 3 (p.98 in my edition) of Spivak's book. The problem is very simple. Let $M$ be a manifold with boundary, and choose a point $p\in\delta(M)$. Now consider an element $v\in T_p M$ which is not spanned by…
Miguel
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Does non vanishing Jacobian implies injectivity?

Let $F:\mathbb{R}^{2}\to \mathbb{R}^{2}$ be a smooth map where its Jacobian $$ Jf(x, y) = \det\begin{pmatrix}\frac{\partial F_{1}}{\partial x} & \frac{\partial{F_{1}}}{\partial y} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial…
Seewoo Lee
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Meaning of $\int_{T_xM} f(z) \ dz$

Question So I'm from an engineering and recently learned about integrating on riemannian manifolds. However, I have been faced with the notation $$\int _{T_xM} f(z) \ dz,$$ where $M$ is a Riemannian manifold and $T_x$ is the tangent space at some…
query
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Integral manifolds of a given distribution.

Given $X_1=\frac{\partial}{\partial x}-yz\frac{\partial}{\partial z}$ and $X_2=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}$ vector fields in $\mathbb{R}^3$, I know the integral curves through $(x_0,y_0,z_0)$ are respectively: $X_1$:…
sheriff
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Negative sign in Leibniz Rule

For $ w \in \bigwedge^j(K) $ and $\mu \in \bigwedge^k(K) $ where $K$ is of dimension $n$, I do understand (or have developed an intuition purely relying on permutations) as to why $$w \wedge \mu = (-1)^{jk}\mu \wedge w$$ However, I am not able to…
mathnoob123
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The points of a segment contained in a surface are parabolic

Let $S\subseteq\mathbb{R}^3$ be a regular surface and let $p$ be a point of $S$. If $p$ lies in a segment contained in $S$ show that $p$ is either parabolic or planar. Well, I think that an idea is to show that the differential of the Gauss map is…
Frankenstein
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I don't understand holonomy well

I'm just trying to understand how a vector can rotate around a smooth loop $\gamma$ on some manifold $M$. By Picard's theorem, the differential equation $\nabla_{\frac{\partial}{\partial t}} W =0$ with initial condition $W_{\gamma(t_0)} = v$ for…
user40276
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Orthogonal coordinate frame

On a Riemannian manifold $(M,g)$, there always exists local orthonormal frame $\{E_i\}$ with respect to the metric $g$. But there does not necessarily exist orthonormal coordinate frame $\{ \frac{\partial}{\partial x^i} \}$ in a small neighborhood…
Dai
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finding a differential curve tangent to a distribution

I have the following distribution on $\mathbb{R}^3$ $${\cal{D}}_{(x,y,z)} = \langle\{\partial_x,\partial_y + x\partial_z\}\rangle$$ I want to show that for any $(x,y,z)$ in $\mathbb{R}^3$, there exists a path $\gamma$ from $0$ to $(x,y,z)$ tangent…
user405156
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Lie derivative is the lie bracket

I am reading the proof of the equivalence between the Lie derivative and the Lie bracket. We define the Lie derivative $\mathcal{L}_X Y$ as $F'(0)$ where $F(t)=\Phi_{{-t}_{*\Phi_t(p)}}(Y_ {{\Phi_t(p)}})$ and $\Phi_t$ is the local flow of $X$ in a…
J. Salieri
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verify k-form is exact

I wonder how to check some simple 1-form or 2-form is exact. For instance, 2-form $w=xdy \wedge dz + ydz \wedge dx − 2zdx \wedge dy$ or 1 form $w=(2x^2y^2+6xy^3)dx + (8x^2y+x^2y^2)dy$. I know that by definition, if there is an $f$ for which…
user53109
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Name of a degenerate foliation

Last summer I went to a talk where the speaker had a specific name for a certain kind of degenerated foliation on a manifold. Sadly, I forgot that name. Question : what is that name ? (Sorry if my question is vague) e.g. If you look at the level…
Noé AC
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Describing all plane curves with constant curvature

I know that by Frenet-Serret, we have (I know this is only for curves parametrized by arclength, but since every plane curve can be reparametrized by arclength, there's no loss of generality): $t'(s) = k(s)n(s) \Rightarrow t''(s) = k(s)n'(s)$ (since…
Matheus Andrade
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Nonexistence of a global coordinate system

I asked this question in https://mathoverflow.net/, but was advised to ask it here. So here it is. I just started a self-study of differential geometry and topology. And in several text I came accross the question, asking to show that the global…
Tomas
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Geodesics through two given points

For me, from the definition of a geodesic: Let $\gamma : t \rightarrow \gamma(t)$, $t \in I$, be a curve in a manifold $M$. The curve $\gamma$ is called a geodesic if the family of tangent vectors $\dot\gamma(t)$ is parallel with respect to…
Hans-Peter Stricker
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