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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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Incompatible first and second fundamental forms

Say the first and second fundamental forms of a surface (a and b) in 2D are incompatible (i.e. they do not satisfy the Codazzi-Mainardi equations), then the "surface" cannot be embedded in 3D. Is this surface embeddable in some (albeit unknown)…
dpholmes
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Rotor of a vector field in coordinates $x^1,x^2,x^3$

Let $U\subset \mathbb{R}^3$ be an open subset endowed with a triple orthogonal coordinate system $\{x^1,x^2,x^3\}$ and let $X$ be a smooth vector field on $U$. The vector field rot$X$ (or curl$X$) which is called a rotor (or curl) of a vector field…
user332350
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Definition of a smooth function between surfaces

If $S_1, S_2 \subset \Bbb R^3$ are two smooth surfaces, then what is the formal definition of a smooth map from $S_1$ to $S_2$? I am studying from Pressley's EDG, and the definition is given only in the case where each of $S_1$ and $S_2$ has an…
user258700
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Confusion about covariant derivative in $\mathbb R^n$

The Levi-civita connection on $\mathbb R^n$ corresponds to the usual directional derivative. In this sense I expect the following to hold: $$ \left(\nabla_{\partial_i}\partial_j\right)f=\partial_{ij}f. $$ On the other hand:…
John
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Is the converse of this true?

I was reading this reference and I was wondering if someone can provide a proof of the converse, well if it is true of course, I mean if two smooth manifolds are diffeomorphic then they have to be homeomorphic, and if fact does the inverse of a…
user162343
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How to best calculate Christoffel symbols of metric $du^2 +g^2(u) dv^2$?

Suppose that the first fundamental form is $du^2+g^2(u)dv^2$. Calculate $\Gamma_{11}^1, \Gamma_{11}^2, \Gamma_{12}^1, \Gamma_{12}^2, \Gamma_{22}^1, \Gamma_{22}^2$. In lectures we've been given 6 formulas for the Christoffel symbols, all of this…
user320757
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An example of the first/second fundamental form

I have some trouble understanding the first/second fundamental form, so I guess a worked-out example would really help. Let's say for the graph of a function $g(x,y)$ with respect to the natural chart. What are the matrices for the first fundamental…
adrw_k
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function on a smooth manifold

Consider a function $f:M\rightarrow\mathbb{R}$, where $M$ is a $C^{\infty}$ manifold. Recall that a function f is smooth if $\forall$ $p$ $\in M$, $\exists$ a smooth chart $(U,\phi)$ that contains $p$ such that $f o \phi^{-1}$ is a smooth map from…
KnobbyWan
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Equivalent condition for the normal lines of a curve $\alpha(s)$ to be equidistant from a fixed point

Let $\alpha : I \rightarrow \mathbb{R^2}$ be a curve parametrized by arc length. Show that all normal lines of $\alpha$ are equidistant from a fixed point if and only if there exist numbers $a,b \in \mathbb{R}$ such that $k(s) = \pm…
salvatore
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Defining a smooth unique envelope function.

Given $a(t), b(t)$ and $c(t)$ defined on $I \to \mathbb{R}$, we want to define the smooth function, i.e. the envelope, $\gamma(t)$, of the family of lines $a(t)x + b(t)y =c(t)$. To find $\gamma(t)$ we solve the system $$ a(t)x + b(t) y = c(t)…
Yuugi
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Question about integral curve on a manifold

In Warner's book on page 36 a curve $\gamma:(a,b)\rightarrow M$ is defined to be an integral curve iff $$d\gamma(\frac{d}{dr}|_t)=X(\gamma(t))$$ Could anyone explain to me the left side in detail and break it into coordinates? Here $X$ is a vector…
Myshkin
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Vector bundle with Moebius strip as base space

This question has a motivation in physics, thus its formulation may not be entirely rigorous. Let $f$ be a function that takes values on a Moebius strip of fixed length $L$ and maps them to operators in a Hilbert space, that is to say the function…
Stan
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Show that curve lies on a sphere

Let $\alpha$ be a curve (in $\mathbb{R^{3}}$) with natural (arc length) parametrization which all osculating planes have exactly one point in common. Show that $\alpha$ is a spherical curve (lies on a sphere).
MasterM
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Difference between geodesics and principal lines of curvature

As much as I understand it, a geodesic line of curvature is a line on the surface such that the projection on the tangent plane of its curvature vector is $0$ at every point. Principal lines of curvature are lines such that their tangent direction…
Mykolas
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The quadric contains the whole line

I am looking at the following exercise: Show that, if a quadric contains three points on a straight line, it contains the whole line. Deduce that, if $L_1$, $L_2$ and $L_3$ are nonintersecting straight lines in $\mathbb{R}^3$, there is a quadric…
Mary Star
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