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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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A question on tangent plane (from Do Carmo)

From 'Do carmo Differential Geometry of curves and surfaces' On page 89, #9. Show that the parametrized surface S given by $$ \text{x}(u,v)=(v\cos{u},v\sin{u},au) $$ Compute its normal vector $N(u, v)$ of a tangent plane of $\text{x}$ at $(u,v)$…
user63310
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Topology on the set of all rank $k$ distributions of a manifold?

Suppose I'm given a manifold $M$ of dimension $n$ and I want to consider the set $$ X = \lbrace H \subset TM | H \textrm{ is a distribution of rank } k \rbrace.$$ What is the usual way to introduce a topology on $X$? I was thinking, take an open…
Matija Sreckovic
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What is going on with the nodal cubic?

Let $y^2 = x^2(x+1)$ be the nodal cubic, where $x,y \in \mathbb{R}$. Parametrize the curve by $x(t) = t^2 - 1$ and $y(t) = t^3 - t$. Then the curve fails to be regular if $\gamma'(t) = \langle x'(t), y'(t) \rangle = 0$, i.e., if $x'(t) = y'(t) =0$.…
user506388
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Missing "trivial step" in the proof of the fundamental theorem of the local theory of curves.

Given $k_o,\tau_o: I \to \mathbb R, \mathcal C^\infty, k_o>0,$ there exists a curve $\alpha:I\to \mathbb R^3$ parameterized by arc length, such that $k(s)=k_o(s)$ and $\tau(s)=\tau_o(s)$ that is unique up to direct isometries of $\mathbb…
Antoni Parellada
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How can we calculate the basis for right invariant vector fields from basis left invariant vectr fields

I want to calculate the right invariant vector fields from left invariant vector fields. The fact I am using is that for a driftless system $\dot X= XA$ we have $\dot Y= -AY$ where $Y=inverse Y$
yasir
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Question regarding the definition of affine connection

In the definition of affine connection, there is $\nabla_X (fY) = \mathrm df(X)Y + f\nabla_XY$ where $X,Y$ are vector fields (or their generalizations) on a smooth manifold $M$ and $f$ is smooth function from a smooth manifold $M$ to $\mathbb{R}$.…
Zeus
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A problem about the definition of regular curve?

This is a standard definition of regular curves. My question comes from the words "In other words". If $c:[a,b]\rightarrow M$ is an immersion, then by definition we have $$c_{*,p}:[a,b]\rightarrow T_pM$$ is injective, which by group theory means…
Charles
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How to calculate the Gaussian curvature of a non-embedded surface

I know how to calculate the Gaussian curvature of an embedded surface using first and second fundamental forms, but how does one calculate the curvature of a non-embedded surface like the hyperbolic plane? Is there some general method to employ? The…
user61496
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How to calculate the Gaussian curvature of a graph of a function

Consider the surface $\{(x,y,F(x,y))\} $ where $F:\mathbb{R^2} \to \mathbb{R}$ is smooth. How would would evaluate the Gaussian curvature at the general point $(x,y,F(x,y))$? I've tried writing out the first fundamental form and using the Brioschi…
user61496
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Curvature gradient from second fundamental form coeffiecients

I have Monge patch surface given as $h=h(u,v)$ and analyze it in points of interest using the usual formulas for coefficients of the first and the second fundamental forms: \begin{align} E&=1+h_u^2, & e&=\frac{h_{uu}}{(1+h_u^2+h_v^2)^{1/2}}, \\ …
eudoxos
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How are geodesics and parallel transports actually defined?

Let $\nabla\colon\Gamma(TM)\times\Gamma(E)\to\Gamma(E),(X,\sigma)\mapsto\nabla_X\sigma$ be a connection on a vector bundle $E$ over a smooth manifold $M$, and let $\gamma\in C^\infty(I,M),I:=[0,1]$ be a smooth curve in $M$. In our lecture we…
Cubi73
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Geometric relationship of the second derivative along a curve on a surface to the normal to the surface at point p

The normal curvature of a curve $C$ on a surface $S$ - please correct - can be estimated as the dot product of the second derivative with respect to arc length with the normal (orthogonal) to the plane at $p:$ $$\kappa_n=\langle C_{ss},\vec…
Antoni Parellada
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Understanding the choice in the horizontal subspace

I am taking a course in differential geometry but my background is in physics. I am struggling to understand how there is an 'choice' in the horizontal subspace when one is considering a vector bundle $\pi : E \rightarrow B$ The vertical subspace at…
Meep
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Transformations law for tensor: why $\partial_{\mu} A^{\nu}$ is not a well defined tensor ? Where is my mistake.

I totally reformulate my question to express precisely my problem. I reason on tensor $(1,1)$ for simplicity even if what I say could be more general. A quantity $T^{\nu}_{\mu}$ i said to be a tensor if under a change of variable $x \rightarrow…
StarBucK
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"diffeomorphic to a $C^1$ manifold"

I'm reading a paper where it is shown a topological manifold $N$ has a $C^1$ structure. The very next concept that is expressed is that prior knowledge of an existing homeomorphism $h: M\rightarrow N$ (where $M$ is $C^\infty$) let us now claim it is…
Mud
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