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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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Finding a smooth function less than some given (positive) continuous function

Let $M$ be a smooth manifold ($dim\ge 1$). Let $f:M\to\mathbb{R}$ be a positive continuous function. Prove there is a smooth map $g\in C^{\infty}(M)$ such that $0
Bey
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The Curve in $R^n$

Let $r:(a,b)\rightarrow{R^n}$ with $|r^{'}|=1$ is a natural parameter curve in $R^n$. If $e_1(s)=r'(s),e_2(s),...,e_n(s)$ form an orthonormal frame, then we have Frenet formulae: $e_{i}^{'}=-k_{i-1}e_{i-1}+k_{i}e_{i+1}$ and…
gaoxinge
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Meaning of restriction of a vector field on a submanifold

I'm trying to make sense of what restricting a vector field means. More specifically, if we have $S^3$ as a submanifold of $\mathbb{R}^4$ and the vector field, say, $X=(x_4,-x_2,x_3,-x_1)$, what does $X\restriction{S^3}$ mean? My confusion lies in…
Weltschmerz
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Where does the invariant expression for the exterior derivative come from?

So I've just spent about four TeXed pages (plus about a dozen TeXed pages of discarded work) proving the identity \begin{align*} d \omega(\zeta_1, \ldots, \zeta_{k+1}) &= \sum_{i=1}^{k+1} (-1)^{i-1} \zeta_i \cdot \omega(\zeta_1, \ldots,…
Daniel McLaury
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Map of constant rank 1

A friend asked me a question which seems to be easily implied by the normal form of constant rank maps but it may be not so obvious. Let $f$ be a smooth map from $\mathbb{R}^2$ to itself, of constant rank $1$, which equals the identity on $\{0\}…
Jeremy Daniel
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condition for curve on a sphere

Let $\alpha (t)$ be a curve such that $|\alpha'(t)|=1$ for all $t\in\mathbb R$. Assume $k(t)\neq 0$, $k'(t)\neq 0$ (whereas $k=|\alpha''(t)|$ is the curvature) and $\tau(s)\neq 0$, whereas $\tau$ is the torsion. Prove: The trace of $\alpha$ lies on…
dinosaur
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Motivation for supermanifolds

Physicists have invented supersymmetry in which they use new variables, mathematically corresponding to Graßmann numbers (elements of some exterior algebra) and physically to "fermionic degrees of freedom". Other notions have been developped as…
Benjamin
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help in understanding tangent vectors

In Aaron's answer here... "Given a manifold $M$, and a point $p\in M$, we have a vector space $T_pM$ of the tangent vectors to $M$ at $p$. For example, if you take the hollow sphere sitting inside $R^3$, you can look at the plane that sits tangent…
Rajesh D
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Jacobian matrix rank and dimension of the image

I am having problems with the following question: What is the relation between the rank of the Jacobian matrix of $f$ (which is continuously differentiable) and the dimension of the image of $f$? Is there a theorem about it?
Alfonso Fajardo
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A good lower bound on the maximum curvature in a loop

Suppose $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ is a $C^\infty$ curve, parameterized by arc length ($\left\|\alpha'(t)\right\| = 1$), and with $\alpha(0) = \alpha(\ell)$. Show that there exists a $t_0 \in [0,\ell]$ such that $ \left\|…
yasmar
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Show that the arc length of a curve is invariant under rigid transformation.

Show that the arc length of a curve is invariant under rigid transformation. The curve here is in $\mathbb R^3$, and the definition of arc length is $\int^b_a||\bf r'$$(t)||dt$. This theorem appears in my book without proof, can somebody please…
JSCB
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Kernel of the tangent map

If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that if $\ker(\mathrm{T}\varphi(x))$ is zero, $\varphi$ is injective in…
mmm
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If all circles have constant geodesic curvature, does the surface have constant curvature?

Let $(S, g)$ be a connected two dimensional smooth manifold which is equipped with a smooth Riemannian metric. Let $\nabla$ be the associated Levi-Civita connection. Consider a smooth unit-speed curve $\gamma$ on $S$ which parameterizes the boundary…
Geoffrey Sangston
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Every manifold admits a vector field with only finitely many zeros

Let $M$ be a smooth manifold. I am trying to prove that $M$ admits a vector field with only finitely many zeros. This will follow if we can find a function $f : M\rightarrow \mathbb R$ such that $df$ has only finitely many zeros, but I cannot find…
user15464
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How to embed Klein Bottle into $R^4$

I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem. The problem is: Show that the mapping $F:\mathbb{R}^2\to\mathbb{R}^4$ given by $$F(x,y)=((r\cos y+a)\cos x,(r\cos y+a)\sin x,r\sin y\cos\frac{x}{2},r\sin…
YYF
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