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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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Variational field on $S^n$

This is my first post/question here, so I hope that I do everything right... I'm currently preparing for an exam and therefore trying to solve this exercise: Let $p\in S^n$ and $v \in T_p S^n$. Compute the geodesic through $p$ with initial…
snom
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Show that an open subset of a smooth manifold is again smooth.

This step is essential in proving that $GL(n,\mathbb{R})$ is a smooth manifold. I already proved that $GL(n,\mathbb{R})$ is an open subset of $M_n(\mathbb{R})$. Can you help me prove the above statement? Thanks! p.s. My idea (but I'm not sure) is…
John Thompson
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Show that the intersection point of the normals converge to a point on the trace of the evolute

Let $\alpha(t): I \to R^2$ be a regular parametrized curve. Assume that $k(t) \neq 0$. The evolute is defined as the curve: $$\beta(t)=\alpha(t)+\frac{1}{k(t)}n(t)$$ Consider the normal lines of $\alpha$ at two neighboring points $t_1,t_2$. Let…
user53970
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Definition of the Tangent Space

I'm watching a series of lectures on differential geometry, and I've run into a bit of a problem with the definition of the tangent space. We first defined a tangent space as $\{(p,v) | v \in \mathbb{R}^n\}$, which makes sense to me: it's the set of…
anjruu
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What happens when the normal to surface is zero?

In Differential Geom we're always given that surfaces should be regular, meaning the partial derivatives at every point are linearly independent, or the normal is non-zero. I get that the tangent space isn't well defined when the partial…
Ted Jh
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Given a regular compact surface $S$ in $\mathbb{R}^3$ proof there exists a line in $\mathbb{R}^3$ which intersects perpendicularly with $S$ twice

I need to proof that given a regular compact surface $S$ in $\mathbb{R}^3$ there exists a line in $\mathbb{R}^3$ which intersects perpendicularly with $S$ twice. Could you help me?
Damaru
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Generalized Laplace--Beltrami operators

Given a smooth surface $M$, we have the well-known Laplace--Beltrami operator which in coordinate-free form is given by $\mathrm{div}_M\,(\mathrm{grad}_M)$. However, we can also consider the more generalized form…
nearly_phd
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Flows of $f$-related vector fields

If we have a smooth map between manifolds $f : M \rightarrow N$ and (say, complete) vector fields $X$ in $M$ and $Y$ in $N$, we say they are $f$-related when $Y(f(x)) = d_x f(X(x))$ for all $x \in M$. I've heard it then claimed that, if $X$ is…
Pedro
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Closed Smooth Curve with Curvature 1 is a Circle

I'm trying to show that a closed, smooth plane curve with curvature 1 is a circle. The Frenet equations are: \begin{align} t' &= kn \\ n' &= -kt - \tau b \\ b' &= \tau n. \end{align} Now, I've shown that if $\alpha(t)$ is a plane curve, then $\tau…
Joshua Bunce
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Diagonalization of Riemannian Metric and the Laplace Beltrami Operator

Consider the local representation of the Laplace Beltrami operator on a Riemannian n - dimensional manifold $(M,g)$: \begin{equation} \triangle_g = \frac{1}{\sqrt{\text{det}(g)}} \sum^n_{i,j = 1} \frac{\partial}{\partial x^i} g^{ij}…
harlekin
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A differentiable map doesn't depend on the parametrization

In Do Carmo's Differential Geometry of Curves and Surfaces there's an excercise in section 2-3 that says: Prove that the definition of a differentiable map between surfaces does not depend on the parametrization chosen. According to the book,…
Ana Galois
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Why compact surfaces can be regarded as region without boundary?

I have been reading DoCarmo and feel quite confused by that he mentioned several times that compact surfaces can be regarded as regions without boundary, which is used in the proof of a corollary of Gauss-Bonnet and several other places. But I can't…
Hui Yu
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Sobolev spaces on manifolds.

Let $(M,g)$ be a compact oriented Riemannian manifold and $E\to M$ be a vector bundle with metric $h$ and a connection $\nabla$. Then one define the sobolev space $W^{k,p}(E)$ as the sets of $L^p$ section $u$ whose weakly covariant differential…
MiGang
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Killing vector field of the sphere

Given a tangent vector field $X(x,y,z) = y\frac{\partial}{\partial x} -x\frac{\partial}{\partial y}$ of the sphere $S^2 \subset \mathbb{R}^3$. Compute the Levi-Civita covariant derivative $\nabla_{v_p}X$ of any tangent vector $v_p$. Secondly, show…
Maria
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When aren't Christoffel symbols symmetrical with respect to bottom indices, and why?

When aren't Christoffel symbols symmetrical with respect to the bottom indices? Why isn't the symmetry of second derivatives true in this case?
Aleksey
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