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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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Why is the Gauss map useful?

Currently I'm studying differential geometry, and more specifically the Gauss map. I'm using the (in)famous Do Carmo Differential Geometry and Surfaces book. I'm having a hard time understanding the use of the Gauss map. I understand what it's…
Maurice
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reconstructing space curves from curvature and torsion

Given $\kappa (s)$ and $\tau (s)$ and a frenet apparatus $\lbrace T_0,N_0,B_0 \rbrace$, how can you reconstruct a space curve? I know I need to use the frenet-serret equations, but I can't put a finger on it. Thanks.
NaiveMathMajor
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Spin structure on Whitney sum

Suppose $E_1$ and $E_2$ are oriented vector bundles with $w_2(E_i)=0$ over a compact manifold $M$. Then $E=E_1 \oplus E_2$ admits a spin structure too. Can we choose a spin structure of $E$ such that the associated spin vector bundle $S(E)$ is…
Wilhelm L.
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Integral of the ratio of torsion and curvature

I want to show that, if every closed curve $\gamma$ on the connected surface $S$ satisfies $$ \int_{\gamma} \left(\frac{\tau}{\kappa}\right)ds =0 $$ where $\tau$ and $\kappa$ are the torsion and the curvature of the curve($\kappa>0$). Then $S$ is a…
Bob brant
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conservation of dot product with parallel transport

I have a question about the parallel transport of a vector : Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle…
user359328
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Nowhere vanishing magnetic helicity

Suppose you are given a nowhere-vanishing exact 2-form $B=dA$ on an open, connected domain $D\subset\mathbb{R}^3$. I'd like to think of $B$ as a magnetic field. Consider the product $H(A)=A\wedge dA$. At least in the plasma physics literature,…
Josh Burby
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Basis of cotangent space

The derivative of a map $F$ between manifolds $M$ and $N$ is defined by $$F_*X(f)= X(f \circ F)$$ where $X \in T_P(M)$, the tanget space at the point $P$. We know that $$\left\{\frac{\partial}{\partial x^i}\bigg|_P\right\}_i$$ is a basis for…
hopo2
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Indices in differential geometry

Often times in differential geometry it is convenient to use Einstein summation notation, and there it is presented to beginning graduates and advanced undergraduates alike that if you see two indices that are the same letter with one upper and the…
Jeff
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Geodesics that self-intersect at finitely many points

Notations $M$ will denote a smooth manifold and $\nabla$ an affine connection on it. A smooth curve $\gamma\colon I \to M$ will be called a geodesic if it is $\nabla$-parallel along itself, that is $\nabla_{\dot{\gamma}(t)}\dot{\gamma}=0$ for every…
Giuseppe Negro
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Basis one-form and basis vector confusion

Still trying to teach myself some basic differential geometry in relation to general relativity. I've read that, in relation to basis vectors $e_{\nu}=\partial_{\nu}$ and basis one-forms $\omega^{\mu}=dx^{\mu}$, then…
Peter4075
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What are half-forms?

Apparently, objects called half-forms exist in differential geometry. If for instance a one form could be written as $d\omega$, a half-form might be denoted $\sqrt{d\omega}$. These objects are very peculiar and I have not been able to find any real…
Kagaratsch
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Definition of vector field along a curve

Let $γ : I→R^3$ be a regular parametrization of a curve C. If asked what a vector field on C is I would perhaps answer like this: 1) "a smooth function $v$ associating to any point $γ(t)$ of C an element of the tangent space of $R^3$ at $γ(t)$".…
user242708
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Why is $\ker\omega$ integrable iff $\omega\wedge d\omega=0$?

Suppose $\omega$ is a nonvanishing $1$-form on a $3$-manifold $M$. It's known that $\ker\omega$ is an integral distribution iff $\omega\wedge d\omega=0$. I'm trying to understand this, but I don't get why $\ker\omega$ integrable implies…
Clara
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What is the difference between abstract index notation and Ricci index notation?

I'm reading Straumann's GR text and he talks about the difference between abstract index notation and Ricci index notation very briefly. So I read the wiki article, but that did not help much. Say we have the Ricci tensor and two vectors. What does…
Ryan Unger
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Are there any compact embedded 2-dimensional surfaces in $\mathbb R^3$ that are also flat?

Let $\overline{g}$ be the flat metric on $\mathbb{R}^3$. I would like to know if there is any compact embedded 2-dimensional surface $M$ in $\mathbb{R}^3$ (without boundary) such that $\iota^*\overline{g}$ is flat, where $\iota: M \hookrightarrow…
user20609
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