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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).

32835 questions
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Integrating a $2$-form over a simplex.

So, given the $2$-form $$\omega=x_1\text{d}x_{2,3}+x_2\text{d}x_{1,3}+x_3\text{d}x_{1,2}$$ and the simplex $$\sigma=[e_1+2e_3,2e_2,e_2-e_1],$$ I defined the function $$g:\mathbb{R}^2\to\mathbb{R}^3$$ $$(x_1,x_2) \mapsto…
user104587
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Normal Coordinates on a Riemannian Manifold

Consider Riemann Normal coordinates on a manifold. Consider a point other than the origin. Given that the metric has vanishing derivatives at this point, is it correct to deduce that the metric is Euclidean at this point? If the deduction is…
Kong
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Non trivial vector bundle

For every line $l$ in $\mathbb{R}^3$ we write $l^\perp$ for the plane orthogonal to $l$. Let $F$ be : $$F = \{(u,l) | l\in\mathbb{P}^2(\mathbb{R}),u\in l^\perp\}$$ How do you show that this is not isomorphic to the trivial bundle on…
Zorba le Grec
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Mean curvature operator in S^n

Consider the sphere $S^n$. By using the stereographic projection we can identify $S^n \setminus N$ with $\mathbb{R}^n$, where $N$ is the North pole of $S^n$. The metric then is given by $\frac{dx^2}{(1+x^2)^2}$. Now we consider the graph of a smooth…
nicolas
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Can you embed a manifold using gradient descent?

Say you have a 2-sphere endowed with a metric. You embed this into $\mathbb{R}^3$. The intrinsic metric won't match the actual distances between neighbouring points in $\mathbb{R}^3$. If you calculated the total error of the mismatch, you might be…
zooby
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Second Fundamental Form, Shape Operator, Curvatures and their relation

Consider a parameterization $r:U\rightarrow M$ for a regular surface $M$.Denote by $n$ the unit normal to the surface. I was given the following defition for the matrix $B$ that defines the second fundamental form: $$B= \begin{pmatrix} …
MasterJ
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Are the level sets of any norm non negatively curved?

It is known the $n$-sphere, the level set of $\|\cdot\|_2$, i.e.$\{x \in \mathbb{R}^{n+1}\, | \| x\|_2 =c \, \}$, has a non negative sectional curvature. I wonder if this level set of any smooth norm $\|\cdot\|$ on $\mathbb{R}^{n+1}$ is also non…
jaogye
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Why can't a curve inside a circle have a point with a smaller curvature than the circle itself?

I can't prove formally this idea which I'm almost 100% sure: If a curve is inside a circle than it can't have any point with a Smaller curvature than the circle itself. How can I prove this simple and intuitive idea formally?
user42912
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How to calculate the tangent space for some concrete examples?

I don't understand how to calculate the tangent space of some concrete examples. For example, can someone explain me how to calculate the tangent space in a point $p$ on the sphere? And how to calculate the tangent space in a point on the torus?
g.pomegranate
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Estimation on $\mid P_{x,x'} \overrightarrow{xy}-\overrightarrow{x'y}\mid$

Let $M$ a relatively compact differentiable manifold equipped by a riemannian metric, and the Levi-Civita connection $\nabla$. We assume that every couple of points in $\mathcal{M}$ are linked by one geodesic of minimal length. Let $x,x',y$ three…
jowe_19
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Extending vector fields along a loop

Let $M$ be a smooth manifold of dimension $n>1$. Let $\ell:=\gamma(S^1)$ be a smooth embedded loop in $M$ for some $\gamma:S^1\to M$. Let $\eta_1, \eta_2\in \mathfrak{X}(\ell;TM)$ be two smooth $TM$-valued vector fields along $\ell$. Question : is…
Noé AC
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pullback of 1 form.

Let $i:S^2 \to \mathbb{R}^3$ be inclusion. Let $\omega=dz$ be a 1-form on $\mathbb{R}^3$. I want to compute $i^* \omega$ and it vainishes exactly two points. Is it right that $i^* \omega=i^*dz=d(i^*z)=d(z \circ i)$? Why does it vanishes at two…
Gobi
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How does the definition of tangent space in terms of equivalence classes of curves relate with the intuitive notion of tangent vectors?

Let $M$ be an $n$-dimensional smooth manifold. One definition of the tangent space of $M$ at a point $p$ is the set of equivalence classes of curves $(\alpha,c)$, where $\alpha$ is a smooth map from a real open interval $J$ to $M$ such that…
D_S
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(Curvature) Why the radius of the circle that gives the best approximation of a curve $\frac{\|T'\|}{\|\gamma '\|}$

Let $\alpha :I\to \mathbb R^2$ a curve. I want the curvature of $\alpha $ at $p=\alpha (u),$ $u\in I$. I therefore want to find the radius of the circle that gives the best approximation of the curve at $p$. I consider $$T(u)=\frac{\alpha…
MSE
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Existence Of a Proper Function on a Manifold

In my homework I am asked to show the existence of a proper function $f:M\rightarrow \mathbb{R}$, and there is a hint which suggests to use a partition of unity. (here proper means that the inverse image of a compact set is compact) I would like…
Or Kedar
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