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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).

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Calculation of Gaussian curvature from first fundamental form

Let $r(u,v): \mathbb R^2 \rightarrow \mathbb R^3$ be a surface in $\mathbb R^3$. I know how to calculate the Gaussian curvature when both the first and the second fundamental forms are given. Also, it's not quite difficult if only the first…
ShinyaSakai
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Geodesic of helicoid

What is the geodesic of the helicoid? $M=\{ (x,y,z)\in \mathbb{R}^3: x\sin z - y\cos z =0\}$ Whose tangent at the point $p = (1,0,0)$ in the line $r = \{(x, y, z) \in \mathbb {R} ^ 3: x = 1, y = z\}$. I tried with the definition, but I don't know…
kEoz
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Why is negative divergence an adjoint of gradient?

In my notes, I have $\langle F, \nabla f\rangle_{L^2(\mathcal{TX})} = \langle \nabla^* F, f\rangle_{L^2(\mathcal{X})} = \langle -\operatorname{div} F, f\rangle_{L^2(\mathcal{X})}$, where $f$ is a scalar field, $F$ is a vector field, $\mathcal X$…
user10024395
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Is the winding number of a simple closed curve always -1, 0 or 1?

Let $\gamma:[a,b]\to \mathbb{R^2}-\{0\}$ be a closed curve of class $C^1$. Define the 1-form $d\theta$ in $\mathbb{R^2}-\{0\}$ as $d\theta=\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ and the winding number of $\gamma$ to be…
Zero
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What is the first fundamental form?

The first fundamental form of a surface $S$ at a point $p$ is "the quadratic form on the tangent plane $S_p$ inherited from the inner product structure of $\mathbb R^3$". At the same time, the first fundamental form is apparently supposed to…
Jack M
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When does a 2D metric have a 3D-surface representation?

Certain non-flat, 2D metrics can be visualized as a 3D surface. The metric for the surface of the unit sphere, $$ds^2 = d\theta^2+\sin^2\theta\,d\phi^2,$$ would be the most familiar example. Others are more esoteric: in Martin's General…
Doubt
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Elementary question in differential geometry

I am trying to learn differential geometry (i.e., teach myself!) So here is a question that came up. For some $h > 0$, consider the cone $C_h = \{ (x,y,z) \; : \; 0 \le z = \sqrt{x^2 + y^2} < h \} \subset \mathbb{R}^3$ endowed with subspace…
passerby51
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Complex structure on cotangent bundle

If $M$ is a complex manifold with complex structure $J$, why does the cotangent bundle of $M$ carry a natural complex structure, and not an almost complex structure. Is that obvious?
Novak Djokovic
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Submanifolds of oriented manifolds

Let $M$ be an $n$-dimensional oriented manifold. Let $f:M\to\mathbb{R}$ be a smooth function. Suppose $c$ is a regular value of $f$ with $f^{-1}(c)$ nonempty. Show that $f^{-1}(c)$ is an oriented regular submanifold of $M$. By constant rank thoerem,…
Xiang Yu
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Dimensions of immersions vs embeddings

Let's say that you have a manifold which you know can be immersed in $\mathbb{R}^n$. Is there a $k$ such that you can say, for sure, that the manifold is embedded in $\mathbb{R}^{n+k}$? I imagine that there is and that this is common knowmedge, but…
R Mary
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Relation between Lie algebras and Lie groups

I am a little confused as to how to compute generally the Lie algebra of a Lie group and viceversa, namely the Lie groups (up to diffeomorphism) having a certain Lie algebra. The way I did this for classical groups such as $O(n)$ or…
Luc
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What exactly is an embedding in differential geometry?

I have seen several definitions of embedding in differential geometry. Let $f:M \to N$ be an injective smooth map between manifolds. Then I have heard that $f$ is an embedding if: $f_*$ is injective, that is, $f$ is an immersion for any $p \in M$…
gj255
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Definition of embedded and immersed curve

What does it mean to say that a curve in $\mathbb{R}^2$ is embedded? I think a curve in $\mathbb{R}^3$ is embedded if it lies on a plane, but what does it mean in 2d? I searched everywhere but I can't find an answer. Also, is there a simple way of…
poe
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famous space curves in geometry history?

For an university assignment I have to visualize some curves in 3 dimensional space. Until now I've implemented Bézier, helix and conical spiral. Could you give me some advice about some famous curves in geometry history?
nkint
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Adding handles to a sphere

I am trying to work through Boothby's An Introduction to Differentiable Manifolds on my own and, embarassingly, have got stuck at the very first chapter. At the end of section 4, chapter 1 (called: Further examples of manifolds: cutting and…
Dactyl
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