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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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Gaussian Curvature K > 0

If M is a surface with Gaussian curvature K > 0, then the curvature of any curve C ⊂ M is everywhere positive. I was reading this in a textbook and I was trying to decide if this was true or not. I am leaning more towards it being false, but am…
MrT
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Tangent of evolute and singed curvature

This is an exercise from differential geometry textbook by Do Carmo. Let $\alpha:I\to \mathbb{R}^2$ be a regular parametrized plain curve (arbitrary parameter), define $n=n(t)$ and $k=k(t)$, where $k$ is the signed curvature. Assume that $k(t)\neq…
John
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Problem with a Do Carmo problem

I'm trying to solve some Do Carmo problems from his book Differential geometry of curves and surfaces. In section 1-3 prob.5.c., we have the curve: $\alpha:(-1,\infty)\rightarrow\Bbb R^2$ given by:…
Ana Galois
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Prove that a straight line is the shortest curve between two points in $R^n$.

Let $p,q∈R^n$ and let $\gamma$ be a curve such that $\gamma(a) = p, \gamma(b) = q$, where $a$ < $b$. (a) Show that, if $\mathbf u$ is a unit vector, then $$\dot\gamma \cdot \mathbf u\leq \|\dot\gamma\|$$ (b) Show that $$(q - p)\cdot\mathbf u ≤…
Emily
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Geometric interpretation of $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$

Let $\alpha(s)$ be a smooth curve parameterised by arc-length and for fixed $r > 0$ define $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$, where $\mathbf{n}(s)$ is the unit normal vector to $\alpha$ at $s$. What is the geometric interpretation of the…
chaffdog
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Doubt with smooth extensions

Let $(x,y)\in\mathbb R^2$ and $M$ a manifold defined by $M=\left\{ (x,y)\in\mathbb R^2\, |\, y^2+x=0 \right\}$. Let $\pi$ be a projection $\pi(x,y)=(x)$. Let $\phi:\mathbb R\to\mathbb R$ be a diffeomorphism. Let me write $\tilde\pi=\pi_M$, then…
PepeToro
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Differential on a product space as sum of differentials

I would like to derive a rule for differentiating maps of the form $h(f(\bullet),g(\bullet))$ on smooth manifolds which is equivalent to partial differentials of multi-variable functions on $\mathbb R$. Let $A,B,C,D$ some smooth manifolds…
Stan
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Length of loxodrome

On a sphere with radius $R$, find the length of a loxodrome which starts at the equator and makes an angle $\gamma$ with all the meridians. (No equations for such a loxodrome are given, and should be derived.)
Hugo Sterls
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Show the torus of revolution has no umbilic points

Let $\mathbb{T}$ denote the torus of revolution with the usual parametrization: $x(u,v) = ( (R + r\cos(u))\cos(v), (R + r\cos(u))\sin(v), r\sin(u) )$ Show that $\mathbb{T}$ has no umbilic points.
Hugo Sterls
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Motivating differential geometry to high school students

What is the best way to motivate and explain what differential geometry to an audience of high school students? Any tips and suggestions are welcomed!
user162074
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Question about Tangent vectors and coordinate change on manifolds

We can describe a vector on a manifold $M$ of dimension $n$ as follows: Let $p:I\rightarrow M$ with $I$ open interval in $\mathbb{R}$ be a curve in $M$. Now look to $p_0=p(0)$. Locally we can find a coordinate system $(U,x_U)$ around $p_0$ such that…
Berk89
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Euler Vector Fields

I have the following question: If $(x^1,\ldots,x^n)$ are the standard coordinates on $R^n$ and $(y^1,\ldots,y^n)$ are other coordinates on $R^n$, how can we show that $\sum x^i\frac{\partial}{\partial x^i} =\sum y^i\frac{\partial}{\partial y^i}$ ?…
Louis
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Application of Christoffel symbol in differential geometry

When self-studying differential geometry, I find my book involves some clumsy, troublesome calculation about Christoffel symbol when proving theorem, which in fact doesn't have the symbols. I wonder if the symbol is actually useful when for doing…
JSCB
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$ \tau = \sqrt{|b'|} $, is this right?

I'm not sure if there is some wrong manipulation in this. This is not homework, just some simple manipulations I've made here that led me to an identity, and I want to know if this is valid. If not, where is wrong? Consider a curve…
Integral
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Proof of the second Bianchi identity

I'm asked to prove the second Bianchi identity: $$\nabla_{[e}R_{ab]c}^{\;\;\;\;d}=0$$ using the fact that: $$(\nabla_a \nabla_b -\nabla_b \nabla_a)\omega_c=R_{abc}^{\;\;\;d}\omega_d$$ For every diff. form $\omega$. I can't. Using that I'm able to…
MyUserIsThis
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