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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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Question about the second fundamental form

I am studying Riemannian geometry and have a question understanding something. I use Do Carmo's book. In the book, a vector field is defined for isometric immersions: for an immersion $$ f:M\rightarrow\bar{M} $$ a local vector field is defined as $$…
xuehy
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understanding of the first fundamental form

The following is an excerpt from do Carmo's Differential Geometry of Curves and Surfaces about the first fundamental form: I don't understand what "without further references to the ambient space ${\Bbb R}^3$" means. When "treating metric…
user9464
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Surface all of whose normals intersect at a point

I am new to differential geometry and encountered difficulty when trying to solve the following problem from Dubrovin's Modern Geometry It's the first problem in exercise 8.4: Find the surface all of whose normals intersect at a point. Intuitively…
To Chin Yu
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mean curvature is trace of second fundamental form?

My understanding was that, from the Weingarten equations, mean curvature $H$ of a surface in $\mathbb{R}^3$ satisfied $$2H = \operatorname{tr}(g^{-1} b),$$ where $g$ is the first fundamental form (metric of the surface) and $b$ is the second…
user7530
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What exactly is a vector bundle isomorphism

Recall that a vector bundle (of rank $n$) is a family of vector spaces $V_x$ of dimension $n$ that is parameterized by a topological space $X$. In addition there is a continuous surjective map $\pi: E \to X$ where $E$ is also a topological space. On…
student
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$d\omega \wedge \omega=0$ implies $d(f\omega)=0$

I would like to prove the following (without the Frobenius Theorem): On $\mathbb{R}^n$, if $\omega$ is nowhere vanishing $1$-form such that $d\omega\wedge\omega=0$ then there exists (at least locally) $f$ a positive function such that…
Paul
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Is the differential $\mathrm{d}\vec{r}$ a sensible mathematical object?

When doing differential geometry, physicists often use $$\mathrm{d}\vec{r} = \mathrm{d}x^i\space\vec{e}_i$$ for many different things. For instance, they define the holonomic basis $\{\vec{e}^{\space\prime}_a\}$ relative to a coordinate system…
TeicDaun
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How can I show that if the second fundamental form of a surface is identically equal to zero, then the surface is a plane?

This is my question: Let P be a plane considered as a surface in 3-space. Show that its second fundamental form is zero. Conversely, show that if the second fundamental form of a surface is identically zero then the surface is a plane. So far I have…
Lucy
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Necessary and sufficient conditions for a helix

I would like to prove that the following two statements are necessary and sufficient conditions that a curve is a helix. I know that a helix is a space curve with the property that the tangent to the curve at every point makes a constant angle with…
user38268
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Show that curvature and torsion are $κ = \frac{\left|γ'∧γ''\right|}{\left|γ'\right|^3}$ and $\tau = \frac{(γ'∧γ'')\cdot γ'''}{\left|γ'∧γ''\right|^2}$

Suppose that $\gamma : I \to \mathbb{R}^3$ is a regular curve (not necessarily parameterised by arc length). How can we show that the curvature and torsion are given respectively by $\kappa = \frac{|\gamma^{\prime}…
Edwardjones Jones
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Induced metric on $S^2$ from pullback of metric on $\mathbb{R}^3$

I'm going over some GR from more of a differential geometry perspective and had a quick question about a simple calculation - my differential geometry background isn't too strong so I apologise if any of the terminology is incorrect, but I'd be…
Eletie
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push forward of vector field

In Gauge Fields, Knots, and Gravity, exercise 18 is the following: Show that if $\phi:M \to N$ we can push forward a vector field $v$ on $M$ to obtain a vector field $\phi_*$ on $N$ satisfying $(\phi_* v)_q = \phi_*(v_p)$, whenever $\phi(p)=q$. I…
nigel
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Principal bundles on 3-manifolds

If G is a simply connected Lie Group then why is every G-bundle over an orientable 3-manifold trivial? (Why is orientability important?)
Student
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Compact surface with Gaussian curvature is positive, negative, and zero

I have a question regarding an exercise of do Carmo, Differential geometry, p. 282: Let $S$ be a regular, compact, orientable surface which is not homeomorphic to a sphere. Prove that there are points on $S$ where the Gaussian curvature is positive,…
Dover87
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Is every smooth surface the level surface of some function?

Let $\Omega$ be an open smooth domain in $\mathbb{R}^n$, then does there exist a smooth function $u$ such that $\{u<0\}=\Omega$ and $\{u=0\}=\partial \Omega$?
student
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