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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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curvature of the boundary of a convex set is positive

Let's consider $J\subset \mathbb R^2$ such that J is convex and such that it's boundary it's a curve $\gamma$. Let's suppose that $\gamma$ is anti-clockwise oriented, let's consider it signed curvature $k_s$. I want to prove the intuitive following…
Miguel
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Differential of a linear map between matrix spaces

This bit of text comes from Lee's Introduction to Smooth Manifolds. I don't see why (8.15) holds. Note first of all that Lee assumes the Einstein summation convention, while I will not in my formulation. I would think that we…
Sha Vuklia
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How do I show this is strictly positive without using matrices.

Suppose $ds^2 = Edu^2 + 2Fdudv + Gdv^2$ is the first fundamental form of some regular surface patch, show $EG - F^2 > 0$ for each point on the surface patch. So technically I could put $E$, $F$, and $G$ into the metric tensor and define a surface…
Lemon
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Can you explain the difference between the sphere $S^5$ and the manifold $S^3 \times S^2$?

If a cube is suspended in mid air with rubber wires inside a hollow glass sphere, its orientations realizes the sphere $S^3$, also called SU(2), which is the the double cover of SO(3). (Is this correct?) If the glass sphere is swimming on water and…
Gina Martelli
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Is there a connection between the construction of $\mathbb{Q}$ and exotic $\mathbb{R}^4$'s

I learned that there exist so-called "exotic" $\mathbb{R}^4$'s. That is, there exist topological spaces which are homeomorphic but not diffeomorphic to $\mathbb{R}^4$. Quite remarkably, it has been proven that $4$ is the only value of $n$ for which…
TheEmptyFunction
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Frame bundle is parallelizable - Kobayashi

Let $M$ be a Riemannian manifold, and let $L(M)$ be the associated frame bundle. At the end of page 40 of Kobayashi's book, as I understand, it is stated that: There exsits $n^2$ connection forms $\omega_j^i$ on $L(M)$ which are nowhere vanishing.…
anonymous67
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Proving $V_{k}(M)=\dfrac{1}{k} \int_{\partial M} \Phi$

If $M$ is a piece-with-boundary of a k-manifold in $\mathbb{R}^{n}$ where $n\geq k$. I want to show that the k-volume $V_{k}(M)$ of $M$ is given by $$V_{k}(M)=\dfrac{1}{k} \int_{\partial M} \Phi$$ The first step from the solution manual of the…
Dewton
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How can I show that the Mercator projection is in the sphere

Define a function $f: \Bbb R×(0,2 \pi) \to \Bbb R^3$ (Mercator projection) by: $$f(u,\theta) = {1 \over {\cosh\,\, u} }\begin{pmatrix}\cos\,\, \theta\\\sin\,\, \theta\\\sinh\,\, u\end{pmatrix} $$ How can I show that $f(u,\theta)$ is in the sphere…
Jhwana
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Different notions of geodesics

Let $M$ be a (without boundary and not necessarly complete) Riemannian manifold. A map $c\colon [a,b]\rightarrow M$ is called geodesic of type A iff $c$ is piecewise smooth, parametrized proportional to arclength and for all $t\in[a,b]$ there exists…
xyz
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Wrapping a Christmas Tree with Even Spacing and Constant Slope

Last week, when I was wrapping strings of beads around my Christmas tree, I initially had the following design in mind: I wanted each pass around the tree to be evenly spaced (e.g. exactly one foot below each part of the string would be another part…
Matthew Milone
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Complex structures on the Iwasawa manifold.

Let $\mathbb{M}=G/\Gamma$ be the Iwasawa manifold, namely, the quotient space of the set of all matrices of the form $$ G= \left\{ (z_1,z_2,z_3) := \begin{pmatrix} 1 & z_1 & z_3 \\ 0 & 1 & z_2 \\ 0 & 0 & 1 \end{pmatrix} : z_1,z_2,z_3 \in…
Jean Billie
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What is the longest, continuous line I can fit within a rectangular box under minimum radius of curvature and line separation conditions?

I have a rectangular area of known dimensions, $x$ and $y$. I want to draw a continuous (i.e. non-crossing) line inside this rectangle that is as long as possible. The radius of curvature at any point on the line can not be less than a value, $r$,…
Alex
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What makes the hairy ball theorem "hard"?

I am currently taking a course on differential geometry, and have been using Spivak as a reference. On page 69 of "A Comprehensive Introduction to Differential Geometry, Vol. I" Spivak says: "It is a well-known (hard) theorem of topology that this…
grumpy
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Differential Geometry: Isometries preserves angles.

The statement is that: If $f:S_1\to S_2$ is an isometry between two regular surfaces and $v_1, v_2$ are two vectors in $T_pS_1$, then the angle between $v_1, v_2$ = the angle between $(df)_p(v_1)$ and $(df)_p(v_2)$. How can we show that by using…
mathasker
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If the principal curvatures of a surface are constant then either part of plane, sphere, or circular cylinder

We get stuck on this homework problem: Prove that if the principal curvatures of a surface $M \subset \mathbb{R}^{3}$ are constant, then $M$ is either part of a plane, a sphere, or a circular cylinder. In the case $k_{1} \neq k_{2}$ assume that…
Xiao Hong
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