Mathematics Stack Exchange
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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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Covector field on the sphere $S^2$ vanishing?

Covector field on the sphere $S^2$ vanishing? There exists a smooth vector field $X$ on $S^2$ that vanishes at exactly one point, for example at the north pole. My idea is the following: Let $\beta:=\{Y_1:=X, Y_2, Y_3\}$ be a basis for…
user31236
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Differentiation on manifold, second derivative

Let $F$ be smooth function on manifold and $v$ vector field. Set $$G(x) := dF_x(v(x)).$$ If $w$ is another vector field, how do I work out $dG_x(w(x))$? I guess should be $$dG_x(w(x)) = d^2F_x(v(x),w(x)),$$ but am not sure how to make formalism…
Konrad
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What is the relationship between the 2nd Fundamental Form and Gaussian Curvature?

I am looking at the proof for this Lemma: If $S \subset \mathbb{R}^3$ is a regular, compact orientable surface, then $S$ has an elliptic point. The proof concludes with stating that since $II_p$ has a fixed sign, then that implies that the…
JJJ
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Connections and differential equations

I was trying to understand the notion of a connection. I have heard in seminars that a connection is more or less a differential equation. I read the definition of Kozsul connection and I am trying to assimilate it. So far I cannot see why a…
Marketa
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Möbius strip as a non-trivial principal bundle

There is a well-known theorem that a principal bundle is trivial if and only if it admits a global section. I'm trying to get a good picture of what this theorem means. The Möbius Strip can be regarded as a principal bundle of $\mathbb{R}$ over…
octopus
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Geometric characterization of critical points of the Gauss map.

Let $\Sigma \subset \mathbb{R}^3$ an oriented surface by Gauss map $N: \Sigma \rightarrow S^2$. How can I find a geometric characterization of critical points of $N$?
Emiliano Inguscio
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Equivalence of definitions of tangent space

For a given manifold $M$ and a point $x \in M$, we can define the tangent space at $x$, $T_xM$ in two ways (more, actually, but I am just concerned about these two for now): 1) Given a chart $(U, \phi)$, where $p \in U$, call two curves $\gamma_1 :…
Tarnation
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Unit Disk Regular Surface?

I am having trouble proving these two problems: 1) is $\{(x,y,z)\in \mathbb{R}|z=0, x^2+y^2\leq1\}$ a regular surface? I say no because the closed unit disk is a closed surface, so we cannot differentiate on the edges. But how can I define a…
malxmusician212
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What is the definition of $dx$

I have just started to study differential forms. I don't yet fully understand the definition of what a differential form is (it's a $p$-times covariant tensor field) but I know that if $U$ is an open subset of a manifold $M$ a differential form…
self-learner
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Geodesics on Compact Manifolds

Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$? I know that this is certainly not true if $M$ is not compact, but I couldn't find a counterexample for the compact…
Sam
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One-to-one mapping from $\mathbb{R}^4$ to $\mathbb{R}^3$

I'm trying to define a mapping from $\mathbb{R}^4$ into $\mathbb{R}^3$ that takes the flat torus to a torus of revolution. Where the flat torus is defined by $x(u,v) = (\cos u, \sin u, \cos v, \sin v)$. And the torus of revolution by $x(u,v) = ( (R…
Phil M.
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Geodesics of this metric

I have to calculate the geodesics of the metric: $$\left(\matrix {1 &0\\0& x^2 }\right)$$ I've been able to derive its equations, which are: $$\ddot x -x\dot y ^2=0$$ $$\ddot y+\frac{2}{x}\dot x\dot y=0$$ It's easy to check that lines of constant…
MyUserIsThis
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Conformal transformation of curvature tensor

I'm asked to calculate the curvature Riemann and Ricci tensors and the curvature scalar of a metric that comes from a conformal transformation from a flat metric, that is, of a metric $g'=\Omega^2g$ where $g$ is flat. Now, I've been able to…
MyUserIsThis
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Does the Hodge star operator commute with 0-forms?

Consider the three differentiable functions $\alpha,\beta,h: \mathbb{R}^2 \to \mathbb{R}$ and the associated 1-forms $d\alpha, d\beta$, with $d$ being the exterior derivative. Let $*$ be the Hodge star operator. Then, is the following relation…
madison54
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On the Lie derivatives along time-dependent vector fields

I was stimulated by this question. On a smooth manifold $M$, an isotopy $\phi:(t,m)\in \mathbb{R}\times M\mapsto\phi_t(m)\in M$, generates the time dependent vector field $X:(t,m)\in\mathbb{R}\times M\mapsto…
agt
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