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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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Computing a Killing vector field from flow

I am given the following manifold $N=\{(x,y)\in \mathbb{R}^2, y>0\}$ with metric: $$ds^2=\frac{dx^2+dy^2}{y^2}$$ There is a suggestion to take $z=x+iy$ and consider the transformations: $$z\to z+c\,, \quad z\to cz\,,\quad z \to…
Joaquin Liniado
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What is a conormal vector to a domain intuitively?

I read that a conormal vector of a domain is a vector that is tangential to the domain and normal to its boundary. If we consider an open disk in $\mathbb{R}^2$ what is a conormal vector at a point on the boundary? I can't picture it at all.
michael_faber
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How does the differential $df$ act on an element of $T_pM$?

Let $f$ be a smooth real valued function on a smooth manifold $M$. The differential of $f$ is the covector field $df$ defined by $$df_p(v) = v(f)$$ where $v \in T_pM$ and where we are now thinking of $v$ as an element of $\operatorname{Der}(M)$.…
user38268
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Do covariant derivatives commute?

Let $M$ be a differentiable manifold, let $f:M\rightarrow\mathbb{R}$ be smooth and let $v,w$ be vector fields in $M$. Must it be true that $\nabla_w(\nabla_vf) = \nabla_v(\nabla_wf)$ where $\nabla$ denotes the covariant derivative? If so I would…
Mathew
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Relating basis vectors at different points in a small neighborhood of a point in a manifold

I'm reading a section [Core Principles of Special and General Relativity by Luscombe] on how the derivative of a basis vector in a manifold is related to connection coefficient. Quoting (the notation $A^{\alpha}_{\beta'}$ means $\partial…
Shirish Kulhari
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Definition of geodesic not as critical point of length $L_\gamma$ [*]

Context of this question: This question follows from a post Decomposition of a function and chain rule. and discusses on something different. Using calculation of variation we can find critical points of a function of a vaiable curve $\gamma$ with…
Charlie Chang
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Tangent space to circle

I guess I am missing something obvious here. I am reading about vector bundles. (What Karoubi calls 'Quasi Vector-Bundles') An example is the sphere, where for every point $X \in S^n$ we choose $E_X$ (the fiber) to be the vector space orthogonal to…
Juan S
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geodesic line on intersecting planes

I have a problem and I don't understand how to solve it, please help! Prove that if two surfaces in $R^3$ intersect along a curve that is geodesic on both surfaces, and the tangent planes to the surfaces at any point of the curve do not coincide (in…
GIFT
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Length of a planar curve

Let $\gamma:[0,L]\rightarrow \mathbb{R}^2$ be a $C^\infty$ curve parameterized by arc length. We suppose that $\gamma$ is a simple closed curve that bounds a bounded domain in $\mathbb{R}^2$. We denote by $\nu(t)$ the inward unit normal at…
Kosh
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Why is mean curvature extrinsic?

I believe that an intrinsic property is dependent on a surface itself (not on how it is parameterized), and an extrinsic property may vary depending on the parameterization. Why is mean curvature extrinsic then? It's a measure of the curvature of a…
David Faux
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How to know if a curve is plane without calculating its torsion

Given $$\alpha(t)=\left(t,\frac{1+t}{t},\frac{1-t^2}{t}\right)$$ I want to know if there is way of knowing if this curve is plane or not without calculating its torsion. I considered the option of trying to know if its contained in a plane. But I…
Jack Talion
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find the torsion and the curvature of this curve... (it's horrible)

Let's consider the following curve: $\varphi(t)=\begin{cases} (t,0,e^{-\frac{1}{t^{2}}}) & t>0\\ (0,0,0) & t=0\\ (t,e^{-\frac{1}{t^{2}}},0) & t<0 \end{cases} $ I have to compute the curvature and the torsión of the curve. Well first of all, I…
Eustass
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Do Carmo: Linear Killing field anti-symmetric?

In Exercise 3.5a of Riemannian Geometry, do Carmo defines a vector field $v$ on $\mathbb{R}^n$ to be linear if it's linear as a map $v\colon \mathbb{R}^n \to\mathbb{R}^n$. He then asks the reader to prove that such a $v$, defined by a matrix $A$, is…
Avi Steiner
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Understand the need of affine connection

I can't understand why textbook says that directional derivative cannot be defined for general manifold, and a separate affine connection is needed. Assume there are two vector fields $X$ and $Y$. In particular, you have a tangent vector at point…
Rui Liu
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Differential as bundle map and pull back bundle

In the wikipedia article about the pushforward, it is stated that if $f: M\to N$ is smooth, then it induces a bundle map $df: TM \to TN$. It is then claimed that equivalently $f_*=df$ is a bundle map from $TM$ to the pullback bundle $f^* TN$. Why is…
Michael
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