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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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Equivalence of parallel transport, connection and covariant derivative

Here by connection I mean the horizontal distribution. I hear about these three notions are equivalent, i.e. given one we can recover another. In the textbook of Riemannian geometry I have read, usually the existence of covariant derivative is…
user330928
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Unit speed reparametrization, Arc-length parametrization, and Orientations.

In the text "O'neill-Elementary differential geometry (2nd edition)", there are following problems. As we know, any unit speed-reparametrized curve can be viewed as an arc-length reparametrizated curve. Since the arc-length reparametrized curve…
Chris kim
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Seeing that the second fundamental form is the orthogonal component of the Laplacian

I have come across the statement a few times that, for a mapping $u:M\to N$ between a Riemannian manifold $(M,g)$ and a submanifold $N$ of Euclidean space $\mathbb{R}^n$, the part of the Laplacian of M orthogonal to the tangent plane of $N$ is given…
Aerinmund Fagelson
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I call this the Punt on the Wine Bottle Theorem.

My grad school days are 30 years behind me, and to my chagrin proving what seems a simple observation draws upon skills which are too badly atrophied to be useful. The inspiration is a wine bottle. Why do wine bottles have "punts" - i.e.…
Pokep
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Are all smooth manifolds the zero locus of a smooth function?

I am reading this answer here explaining the difference between varieties and smooth manifolds and it says: "...Varieties are cut out in their ambient (affine or projective) space as the zero loci of polynomial functions, rather than simply as the…
student
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surface lying on one side of another surface

I was studying the strong maximum principle for minimal surfaces and came across the statement that surface A lies on one side of the surface B. Can you please tell me what does it mean mathematically? The Theorem Statement is: "If $\Sigma_1…
inquisitive
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To show $X$ is a complete vector field on $M$

Well, I have solved myself the problem : every smooth vector field on a compact manifold is complete. Now I have got this problem which I am not able to progress: let $X$ is a vector field on $M$, suppose $\exists \epsilon >0\ni (-\epsilon,\epsilon)…
Myshkin
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smooth maps between smooth manifolds - Jacobian coordinate independence

I have question about comment in Lee's Introduction to Smooth Manifolds - page 51. Given smooth map $F:M\to N$ between smooth manifolds $M$ and $N$ we say that the total derivative of $F$ at $p\in M$ (given chart $(U,\varphi)$ around $p$ and…
dmm
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A question my proof about the line of curvature

I am working on Exercise 2.4.4 of Differential Geometry and Its Application. The problem statement and my work is available at this link. At the end of my proof, I claimed that $ S_p(\alpha') $ is both on the plane and the tangent plane, so it must…
Andrew Au
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Unit Normal vs Principal Normal

Here is the problem I am working on: Deduce the equation of the main normal and binormal to the curve: $x=t, y=t^2, z=t^3, t=1.$ I remember from Calc-3 that the binormal is unit tangent $\times$ unit normal, and that unit normal is tangent prime…
Heather Madsen
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Hopf Invariant of $f$

To set up notation: Let $f:S^3\to S^2$. For a volume form $\omega$ on $S^2$, $f^*\omega$ is a closed two form on $S^3$, which can be written as $d\alpha$ for some 1 form $\alpha$. The number $$\int_{S^3}\alpha\wedge d\alpha$$ is called the Hopf…
TomGrubb
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Lie Bracket is not a tensor

Suppose that we have two vector fields $V,W \in TM$ with $M$ a differentiable manifold. We define the Lie Bracket as $[V,W](f)=V(W(f))-W(V(f))$ and it's easy to show that this bracket is indeed a vector field. I feel like I'm missing something but…
Abellan
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Wedge product of 1-Forms

I'm trying to write down the wedge product of 2 1-forms on an n-dimensional Manifold. $\alpha = \alpha_1 dx^1 + \alpha_2 dx^2 + \cdots + \alpha_n dx^n$ and $\beta = \beta_1 dx^1 + \beta_2 dx^2 + \cdots + \beta_n dx^n$ I know how to do this for the 2…
Tenak Singh
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Identification of $T_v(T_pM)$ with $T_pM$

In some passages of Do Carmo's Riemannian geometry book he identify $T_v(T_pM)$ with $T_pM$, my question: How one see $T_pM$ as a manifold? who is the atlas? What is the expression of a vector $x \in T_v(T_pM)$ in local coordinates?
Jr.
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Invariant tensors derived from an affine connection

Given $\nabla$,an affine connection in the tangent bundle of a manifold $M,$ we have two very important and cannonical tensors asociated to $\nabla,$ namely the torsion tensor $T$ and the curvature tensor $R.$ Are there any other tensors derived…
Mike Cocos
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