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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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Lie Derivative Equals to Lie Bracket

I am reading the book Introduction to Smooth Manifold written by M.Lee. I am confusing with the concept of Lie derivative. We have $\mathcal{L}_XY=[X,Y]$. However we have $D_XY=X(Y^i)\frac{\partial}{\partial x^i}$ in Euclidean space, and…
gaoxinge
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Killing Field on a Riemannian Manifold

Do there exist a nontrivial Killing field on each riemannian manifold? A Killing field is a vector field whose flow acts on the manifold by isometry.
Hesam
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About metric and the Ricci curvatrue

Recently, I met a question about the relation between $g$ and $-Ric_g$ on the Riemmannian manifold $(M^n, g)$. One said that "without loss of generality that by scaling $g$ in space we have $g \geq - Ric_g$". Is there anyone who can explain it?…
Weeson Dorne
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Differential geometry: $|\alpha(t)|=c\ne 0 \iff \langle \alpha(t),\alpha'(t)\rangle = 0,\forall t\in I$

Let $\alpha: I\to \Bbb R^3$ be a regular para curve. I want to prove that: $|\alpha(t)|=c\ne 0 \iff \langle \alpha(t),\alpha'(t)\rangle = 0,\forall t\in I$ Now $|\alpha(t)|=c\ne 0$ means that this traces some subset or an entire path on the surface…
Differential Geometry
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The diffeomorphism of $\mathbb R^n$

If $f$ is a diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that: (1)$f=\tilde f$ on a neighborhood of $K$. (2)There is a bounded set $V$ and $\tilde…
Summer
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Calculating mean and Gaussian curvature

I am stuck on this question from a tutorial sheet I am going through. Compute the mean and Gaussian curvature of a surface in $\mathbb{R}^3$ that is given by $z=f(x)+g(y)$ for some good functions $f(x)$ and $g(y)$. I tried calculating the first and…
09867
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Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, then every integral curve of $X$ that intersect…
Ervin
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a curve does not need to be injective?

In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective? If it is not, then there might be a two different tangent vector at a point in the euclidean space...which…
Mathcho
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Congruence of two curves with an arbitrary speed?

I'm studying the book "elementary differential geometry" by o'neil. There is a collorary which states that if two curves a(t), b(t) which is defined in the same real line interval has the same speed, curvature, torsion(torsion may differ by sign),…
Mathcho
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question from do carmo diff. geometry

I am studying differential geometry myself from Do carmo and i didn't understand the question : show that if a surface is tangent to a plane along a curve , then the points of this curve are either parabolic or planar . At the question i didn't…
bytrz
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Prove using an admissible unit speed curve and a Frenet frame

Assume $f:(a,b) \to \mathbb R^3$ is an admissible unit speed curve (hence $f^{\prime} \times f^{\prime\prime}$ is never zero) If $f$ lies on the sphere with center $a$ and radius $r$ prove that $f = a - (1/\kappa) \mathbf N - (1/\kappa)^{\prime}…
JimJones
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How to show the following vector bundles are equivalent?

Given a smooth sub-manifold $X$ of $\mathbb{R^n}$ and define the diagonal in $X \times X$ to be $$\triangle = \{(x,x) \mid x \in X \} \subset \mathbb{R^n}\times \mathbb{R^n}$$ and normal bundle to $\triangle$ is defined to be $$N(\triangle)=\{(y,w)…
SamC
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Finding the evolute of a parabola

I previously tried to find the evolute of a parabola by using parameterisation by arc length. It didn't work. While I was hoping for an answer I kept working on it and came up with the following method (unfortunately, something is not quite right as…
student
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Points of $4$-contact of an ellipse and a circle

Consider an ellipse $x^2 + 4y^2 = 4$ given in parametrised form $(2 \cos t, \sin t)$. At a given point $p_0 = (2 \cos t_0, \sin t_0)$ we want to measure how round the ellipse is (i.e. how similar to a circle it is). To do this, let $C(x,y) = (x-a)^2…
student
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Finding integral submanifold passing through the origin

I'm having a little trouble with this problem for Lee - Introduction to Smooth Manifolds (2nd ed). The problem is as follows (Problem 19-5): Let $D$ be the distribution of $\mathbb{R}^3$ spanned by \begin{align*} X&=\frac{\partial}{\partial…
Blake
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