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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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Differential geometry: $S_1,S_2$ are regular surfaces, $f:S_1\to\mathbb{R}^{3}$ is smooth, then $(df)_p (T_pS_1) \subseteq T_f(p)S_2$.

If $S_1$ and $S_2$ are two regular surfaces and $f: S_1\to\mathbb{R}^{3}$ is smooth function and $f(S_1)\subseteq S_2$ and $p$ is a point in $S_1$, then $(df)_p (T_pS_1) \subseteq T_{f(p)}S_2$. How can we prove that implication?
mathasker
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What is a monotonic family of surfaces?

I am following a paper by Eberhard Hopf titled On S. Bernstein's Theorem on Surfaces z(x, y) of Nonpositive Curvature. Not long into the proof for his first lemma, he makes a construction of spheres that satisfy certain properties. Then, he mentions…
D. Brito
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$S^n \times S^m$ submanifold of $2S^{m+n+1}$

I would like to show that $S^n \times S^m$ is a submanifold of $2S^{m+n+1}$. With $2S^{m+n+1}$ we denote the $m+n+1$ sphere with radius $\sqrt{2}$. Here $S^n$ and $S^m$ are the unit spheres. I have trouble with proving this, can somebody help me?…
AL123
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Prove that function from $S^n \times S^m \to 2S^{m+n+1}$ is immersion

I would like to prove that the map $f: S^n \times S^m \to 2S^{m+n+1}: ((x_1,..,x_{n+1}), (y_1,...,y_{m+1})) \to (x_1,...,x_{n+1},y_1,...,y_{m+1})$ is an imersion. Here $2S^{m+n+1}$ is the $m+n+1$ dimensional sphere with radius $\sqrt2$. I know…
AL123
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Sectional curvature and Gauss curvature

Several authors (also of standard RiemGeo books) write that the sectional curvature of a plane $\pi$ contained in $T_pM$, where $(M,g)$ is a Riemannian manifold of any dimension, is the "Gaussian curvature" in $p$ of the surface $S$ generated by the…
Carlo Mantegazza
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I don't understand the proof of following exercise from Do Carmo's differential geometry.

Let me writh down what I'm trying to prove. It comes from Do carmo's differential geometry sec 1.7. Lets $\alpha(s), s \in [0, l]$ be a closed convex plane curve positively oriented. The curve $\beta(s)=\alpha(s)-rn(s)$, where $r$ is a positive…
glimpser
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an example of a curve such that...

Given differentiable functions $k(s),\tau(s)$ with domain $s\in (a,b)$, there exist a differentiable function $\gamma:(a,b)\to \mathbb R^3$ such that $k,\tau$ are it's respectively curvature and torsión. My question: I need an example of a pair…
Eustass
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$V$, $W$ two vector fields on surface $M$, $S$ shape operator, show that $S(V) \cdot W=\nabla_{V} W \cdot U$

This is homework, and we are stuck. Let $V$, $W$ be two vector fields on a surface $M$ (with assumed ambient space $\mathbb{R}^3$). Prove that if $S$ is the shape operator on $M$ corresponding to a given unit normal vector field $U$,…
runeblaze
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Winding number of vector field on surface

I found a term "winding number of vector field with respect to another vector field" in a paper without definition. Because my paper I am reading is talking about the surface, so I don't know if I can use the definition of winding number of vector…
mapping
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Given a $1$ form $\theta$ on $\mathbb{R}^3$ find a 2-dimensional submanifold $M$ such that $\theta|_{M}=0$

I am asked to find a $2-$dimensional submaifold of $\mathbb{R}^3$ such that the $1$ form $$\theta=(z^2+2xy)\cdot dx+(x^2+2yz)\cdot dy+(y^2+2zx)\cdot dx$$ restriced to the manifold $M$ equals zero. My idea is to find $M$ trough its tangent space.…
esteban
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How's the first fundamental form of a surface parameterization with a diffeomorphism?

Let $\sigma: U\subset\Bbb{R^2}\to V\subset S$ a parameterization of a surface $S$, and let $g:\tilde{U}\to U$ be a diffeomorphism between open sets of $\Bbb{R^2}$. I need to obtain a formula for the coefficients of the first fundamental form…
Armando Rosas
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question about geodesic curvature of geometry

I have questions. Can anyone help me to get the idea or figure out this problem? What is the formula of geodesic curvature and what is the easy formula to compute geodesic curvature for any surface? Thanks!
leena adam
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baby do Carmo loxodromes

I am struggling with a problem in baby do Carmo's Differential Geometry of Curves and Surfaces (Problem 2 in Section 2.5), which says: Let $X(\phi, \theta) = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$ be a parametrization of the…
DrHAL
  • 866
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Push-forward of exponential map

I am studying Riemannian geometry, and there is something stated in my lecture notes that I simply cannot understand: If $\gamma$ is a geodesic curve on a manifold M so $\gamma(p)=0$, $\gamma(q)=1$, then we have…
Christine
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How to find the kernel of this distribution?

Consider the linear functional $\omega \in (\mathbb{R}^3)^{*}$, gived by $ω(x,y,z)=dy+y⋅dx$ (The base ${dx,dy,dz}$ for $(\mathbb{R}^3)^{*}$ is dual in the canonical base of $\mathbb{R}^3$ $\{ e_1,e_2,e_3 \})$.The $2$-distribution…
sango
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