Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

Covariant derivative and surface gradient

The surface gradient of a function defined on a surface $\Gamma \subset \mathbb{R}^n$ is defined $$\nabla_{\Gamma} f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal on $\Gamma.$ How do I obtain this from the covariant derivative…
hopo2
  • 433
3
votes
2 answers

How can I prove that a curve with constant nonzero curvature and torsion is a helix?

I've gotten as far as $\mathbf{n''}+(\kappa^2+\tau^2)\mathbf{n} = 0$, which suggests or at least permits trigonometric expressions for every component of $\mathbf{n}$ -- not something that seems to lead to a standard expression for a helix. Am I…
Shay Guy
  • 203
3
votes
2 answers

Approximation of curvature of arbitrary plane curve

Let $S$ be the area (if finite) of the region bounded by a plane curve and a straight line parallel to the tangent line to the curve at a point on the curve, and at a distance $h$ from it. Express $\lim_{h \rightarrow 0} (S^2/h^3)$ in terms of the…
Kelvin Hsu
  • 137
3
votes
2 answers

Definition of Frenet frame for curves in $\mathbb{R}^n$.

I have a question about the definition of Frenet frame here (pg 6) https://www.math.cuhk.edu.hk/~martinli/teaching/4030lectures.pdf . The definition above is summarized below: We are given a regular curve, $c$, in $\mathbb{R}^n$ parametrized by arc…
Arin Chaudhuri
  • 6,349
3
votes
1 answer

Some confusion about normal vector, curvature and normal curvature in Do Carmo's textbook.

There is this part of a text from Do Carmo's Differential Geometry that I don't quite understand. I understand the definitions of curvature, normal curvature, normal section etc. But what I am confused at is the part that says "In a neighbourhood…
user338393
  • 1,165
3
votes
0 answers

Neighborhood about a simple closed curve, and changing coordinated for one forms

I've been struggling with the following two problems for while now, and i am grateful for any assistance or hint to solve them let $\gamma$ be a simple smooth closed curve in R^2 of length L which is parametrized by arclength. Let $N_\epsilon$…
Jessica
  • 31
3
votes
1 answer

The Concept of Isometry under Riemannian Metric's Context

While reading the book Modern Geometry — Methods and Applications Part I, I have a problem reconciling the definition of isometries with the usual version (an isometry preserves the distance between two points). Quoted from the book, an isometry (or…
Kelvin Hsu
  • 137
3
votes
1 answer

Second fundamental form and Weingarten map

I know two definitions of the 2nd fundamental form on 2-surfaces in $\mathbb R^3$: 1) For a parametrization $X(u,v)$ of the surface and the normal vector $\nu$, the 2nd fundamental form is given by the matrix $$ \left( \begin{array} & X_{uu}…
Andy Miles
  • 911
3
votes
1 answer

Showing that a curve $\Gamma^*$ is spherical

I'm having troubles with this differential geometry problem: Let $\Gamma : \overrightarrow{x} = \overrightarrow{x}(s)$ be a curve of $E^3$ with natural parameter $s$ which does not pass through the origin $O$. Now consider the curve $$\Gamma^* :…
user347616
3
votes
1 answer

Local equation for the manifold from its principal curvatures

If $k_1,k_2,...,k_n$ are principal curvatures of a hypersurface in $\mathbb{R}^{n+1}$ then one can apparently locally parametrize the manifold as $(x_1,x_2,...,x_n,y)$ such that, $y = \frac{1}{2}(k_1 x_1^2 +...+k_n x_n^2) + O(\vert x \vert ^3) $ I…
Student
  • 2,755
3
votes
1 answer

Differential on a manifold

I got confused with the differential of a map on a manifold. I try to wrap up what I learned and what I am confused about. Let $X$ be a manifold and $c_1,c_2: (-\epsilon, \epsilon) \to X$ two curves satisfying $c_1(0) = x = c_2(0)$ for an $x \in X$.…
JDoe
  • 704
3
votes
3 answers

Does a differential manifold implies existence of unique tangent space at every point?

I like differential geometry and I want to know if a differentiable manifold implies unique tangent space at every point. I have searched but the definition I have found of differential manifold is that differentiable manifold is a topological…
Mizi
  • 31
3
votes
0 answers

Principal null Direction

I need to understand what the principal null dierctions are in mathematics. Physicists define a principal null direction in a spacetime as a null vector which satisfies the Penrose-Debever equation. I would like to know what one exactly means by…
Masoud
  • 31
3
votes
1 answer

Find surface in $\mathbb{R}^3$ with certain tangent spaces

By Frobenius Theorem, in $\mathbb{R}^3$ there exists a smooth surface whose tangent space is spanned by the vector fields $V(x,y,z)=(x^2+y^2,0,-y)$ and $W(x,y,z)=(0,x^2+y^2,x)$. How can I find this surface? Is there in general a way to find it when…
abrax
  • 261
3
votes
2 answers

How to find the maximal integral submanifold in a concrete case?

Take $M=\mathbb{R}^3$ be a smooth manifold. Consider a distribution $\Delta_{(x,y,z)} = Span\{y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}, z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z} \}$. 1) Show that the distribution…
Honghao
  • 983
1 2 3
…
99 100