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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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Prove that $\frac{d^2 t}{d s^2} = - \frac{\alpha ' (t) \cdot \alpha '' (t)}{|| \alpha ' (t) ||^4}$

I have to prove that if $\alpha(t)$ is a regular curve in space and $\beta(s)$ is its reparametrization of unit speed then $\frac{d^2 t}{d s^2} = - \frac{\alpha ' (t) \cdot \alpha '' (t)}{|| \alpha ' (t) ||^4}$. Attempt: I know that the RHS…
davidaap
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Show that, if every tangent line of a unit speed curve $\beta$ passes through a fixed point $p$, then $β$ is a line.

Just wondering whether my proof is valid or not, feedback would be very appreciated! If the tangent line $T$ always intersects a particular point $p$ then it must be the case that $T$ is constantly pointing either directly towards $p$ or directly…
talrefae
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Normal Curvature Along a non arclength-parameterized curve

I consider the surface $S=\{(x,y,z) \in \mathbb{R}^3: \, z=x^2+y^2\}$. Clearly, it may be parameterizated by $\phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t \to \phi(t^2,t)$. Which is the normal curvature along the previous curve? My problem…
TheWanderer
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Rectifying plane

I'm having trouble understanding this question. If $r(s)$ is a unit speed curve with $\kappa > 0$ and it lies in its rectifying plane for all $s$, then $r \cdot T = s + c$ for some constant $c$ and $ r\cdot B$ is a non-zero constant and the torsion…
Luke Xu
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Geodesic curvature/Metrics in R²

I'd like to have a clear idea on how to solve this (seems to be very simple, but I haven't seen something like that yet). Consider $\mathbb{R}^{2}$ with the metric given by $ds²$ = $\frac{4(dx² + dy²)}{1+x²+y²}$. Compute the geodesic curvatures of…
Br09
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Calculation of a Lie Bracket to show that two basis vectors are a noncoordinate basis

I have unit vector fields defined by $\hat{r} = cos(\theta)\hat{x} + sin(\theta)\hat{y}$ $\hat{\theta} = -sin(\theta)\hat{x} + cos(\theta)\hat{y}$ where $\hat{x} = \frac{\partial}{\partial x}$ and $\hat{y} = \frac{\partial}{\partial y}$ Now to show…
user570877
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Difeomorphisms and boundary conditions

So I asked on physics.stackexchange, but got no answer, so I'll try here: I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism which preserve the boundary conditions in the same…
dingo_d
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Is surface regularity preserved under diffeomorphisms?

I'm studying Do Carmo's differential geometry of curves and surfaces book and I have come up with the following question: Let $S_1 \subset \mathbb{R}^3$ a regular surface and $S_2 \subset \mathbb{R}^3$ such that there exists a diffeomorphism $f:S_1…
Yagger
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curvature of a general curve

Let $\gamma : I \to \mathbb{R}^2$ be a smooth immersed curve, i.e. $\mathcal{C}^\infty$ and $\frac{d\gamma}{dt}\neq 0$ on $I$. I have found the following formula for the curvature $\kappa$ of $\gamma$, but I haven't seen a proof. Does anyone know…
alexlo
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tangential and normal projection of a vector in the ambient vector field of a sphere

I'm having unexpected trouble to perform this computation: Let $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=3\}$ and $v_p = (1,0,0)_{(1,1,1)}$ be a vector from the ambient vector field on $M$. How do I now compute the projections $v_p^T$ of $v_p$ to the…
alexlo
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Basic question about the Gauss map and its relation to unit spheres

I'm studying elementary differential geometry and I'm trying to understand how the Gauss map is a map into the unit sphere. The definition I'm working with is $\textbf{Def}$: Let $S$ be a regular surface. A Gauss map of $S$ is a continous map…
user119264
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Why is the tangent space to a point $p$ of $D\subset \mathbb{R}^n$ isomorphic to $\mathbb{R}^n$

I'm having problems with understanding why is it the case. Suppose $D\subset\mathbb{R}^n$ is an open, connected subset and for $p\in D$ define the tangent space $T_pD$ to be the set of the velocities of all curves $(-\epsilon,\epsilon)\to D$ which…
Jimmy R
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An identity in terms of the Laplace-Beltrami operator

Let $\Gamma:\eta=(\eta_1(x_1,x_2),\eta_2(x_1,x_2),\eta_3(x_1,x_2))$ be a smooth surface in $\mathbb{R}^3$ with induced megetric $g=(g_{\alpha,\beta})$, where $g_{\alpha, \beta}=\partial _{\alpha}\eta\cdot \partial _{ \beta}\eta $. And let $…
user562244
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definition of differentiability on a regular surface

all. I am studying the book "Differential Geometry of Curves and Surfaces" written by do Carmo, and there is one thing that confuses me so much: In his book, a regular surface refers to a subset of $\mathbb{R}^3$ that satisfy certain conditions.…
Boar
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A question from differential geometry

I read a statement in a paper by J. Manton says that: Consider a circle. It is a manifold. It locally looks like $\mathbb{R}$ whereas globally does not. Mathematically, the circle cannot look like an open subset of $\mathbb{R}$; in particular,…
Amin
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