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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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Discrete Holonomy: Tapp's Differential Geometry book. Exercise 6.6

Here is the problem: Let $S$ be a complete oriented regular surface and $\gamma : [a, b] \rightarrow S$ a regular curve in $S$. Let $a = t_0 < t_1 <... < t_n = b$ be a regular partition of $[a, b]$, which means that $t_i = a + i \Delta t$, where…
Plemath
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Representing a Riemannian metric in $\mathbb R^3$ restricted to the upper half of $S^2$

Consider $M := \mathbb R^3$ as a smooth manifold with a Riemannian metric $g := \sum_{i=1}^3 dx^i\otimes dx^i$, where $(x^1, x^2, x^3)$ is the standard coordinate of $M$. Let $N\subset M$ be a submanifold defined as $S^2 \cap H$, where $H = \{(x^1,…
Pteromys
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Showing that every smooth embedded Torus has a point with negative curvature

I'm having a bit of trouble formally proving the following question from differential geometry: Show that every smooth embedded torus in $\mathbb R^3$ has a point with negative curvature. I know intuitively that this is true. If we look at the…
richie
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Simple closed curve $n(s)$ divides $S^2$ into region of equal area

This is from Kristopher Tapp's "Differential Geometry of Curves and Surfaces", Exercise 6.5 titled "Jacobi's Theorem" Exercise 6.5 (Jacobi's Theorem). Let $\gamma:[a,b]\to \mathbb{R}^3$ be a simple closed unit-speed space curve with…
lecdabster
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Tangent map of the inclusion map of a submanifold

Let $M$ be the Minkowski spacetime, let $f\in C^{\infty}(M)$ be defined as $f(m)=x^{0}(m)$, with $\{x^{\mu}\}$ being a global Cartesian coordinates system, and let $M\supset F_{t}=f^{-1}(t)$ be the submanifold relative to a regular value…
fmc2
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Why is there more emphasis on the exterior algebra of covectors than vectors?

Disclaimer: physicist learning differential geometry here. Possible I get a lot wrong, please bear with me. In basic differential geometry it seems there is an emphasis on the exterior algebra of covectors. for example $dx\wedge dy \wedge dz$ is an…
Jagerber48
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Look for an example of a surface which is neither a closed subset of $\mathbb{R}^{3}$ nor a proper subset of a larger connected surface.

This is the problem from Curves and Surfaces(Montiel, Ros). The hint suggests the surface $S$ parametrized as $$X(u,v) = (e^{-u}cos(u), e^{-u}sin(u), v)$$ to be the answer. Though I understand that the surface is not a closed subset of…
MMH..
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What is the relation between the Chern-Gauss-Bonnet theorem and the Hopf index theorem?

I'm reading a paper where the following form of Chern-Gauss-Bonnet theorem is proved: Let $R$ denote the Riemannian curvature tensor associated to the Levi-Civita connection (the Riemannian connection) on a closed Riemannian manifold $X$ of…
Shana
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Is it possible to construct a vector field on a manifold from a curve

Let $M$ be a smooth manifold of dimension $n$. Suppose $X \in \mathfrak{X}(M)$ and $p \in M$. Then, we may find a curve $\gamma:(-\epsilon, \epsilon) \rightarrow M$ where $\gamma(0) = p$ and $\gamma'(0) = X(p)$. I want the other way around: suppose…
James C
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Show there is a line through the origin in $\mathbb{R}^4$ disjoint from a set.

I found this question on an old qualifying exam, but I don't know what ideas and/or theorems I should use to approach it. I tried to use $\mathbb{RP}^4$, but that is the only idea I have for the following: Let $f:S^2 \to \mathbb{R}^4$ be a smooth…
phenomenalwoman4
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Describing the tangent space of a regular surface

In my class, we defined the tangent space $T_pS$ of a regular curve $S$ at a point $p\in S$ as the set $\gamma'(0)$ such that $\gamma$ is a regular curve in $S$ and $\gamma(0)=p$. My particular problem involves a torus $S$ parametrized as…
hello4649
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A question on geodesics on a cylinder

I have solved an exercise on gedeosics on a cylinder but I have a question that makes me doubt the correctness. My strategy to solve this was as follows: Determine and solve the geodesic equations. Show that the solutions are parametrizations of…
Polymorph
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Gaussian curvature invariant under an isometry

This question concerns the transformation of one surface into another. I note the relevant postings but lack the knowledge/expertise to use them. What prior concepts do I need to grasp to understand on an intuitive level why Gaussian curvature is…
Paul Stephenson
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Lie derivative of a one form in $S^1$

I'm reading section 4 in the article Cohomologie équivariante et Théorème de Stokes, it says the following: The circle $S^1$ is parametrized by the angle $\theta$. The action of $S^1$ on $S^1$ is induced by the vector field…
Mira
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Differential of restriction vs. restriction of differential

Perhaps an easy question, but I am not seeing it. Let $\phi:M\to M'$ be a smooth map, and $N\subset M$ a closed submanifold. For $p\in N$, does the map $d(\phi|_N)_p:T_pN \to T_{\phi(p)}M'$ coincide with the restriction of $d\phi_p$ to $T_pN\subset…
Oliver
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