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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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Standard Coordinate Charts On A Sphere

Below are excerpts from Lee's Introduction To Smooth Manifolds for the context of my question: What I am confused about is the part where he talks about $\phi_i^+ \circ (\phi_i^-)^{-1} = \phi_i^- \circ (\phi_i^+)^{-1} = Id_{\mathbb{B}}$. There…
Frederic Chopin
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Existence of Commuting Vector Fields in a Nonintegrable Distribution

Let $M$ be a (smooth) manifold. Given tangent vectors $X_q,Y_q \in T_qM$ ($q \in M$), there exist (locally about $q$) vector fields $X,Y \in \Gamma(TM)$ extending $X_q$, $Y_q$ (i.e., $X(q) = X_q$ and $Y(q) = Y_q$), and such that $[X,Y] = 0$. My…
Dennis
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a doubt on manifold with boundary, critical point, space of jets etc

could any one explain me the following paragraph by a simple example? "a manifold with boundary is understood to be a smooth (real or complex) manifold with a fixed smooth hypersurface. Two functions on a manifold with boundary are called equivalent…
Myshkin
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Local Convexity and Curvature (Do Carmo)

I find myself unable to start the following problem in Differential Geometry of Curves and Surfaces by Do Carmo, Section 3.3 Problem 24.a Edit: (Definition) (Local Convexity and Curvature). A surface $S \subset R^3$ is locally convex at a point p ∈…
Ecotistician
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Second fundamental theorem of calculus on the unit sphere

Without being so sure, the second fundamental theorem of calculus can be written in the following form: Let $f \in {\cal C}^1(\mathbb{R}^n)$. Then, for all $x, y \in \mathbb{R}^n$, we have \begin{align} f(y) = f(x) + \int_0^1 \langle \nabla f(x…
thmusic
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Does the tangent bundle on a 2-sphere span $\mathbb R^3$ and how are the operations defined?

As a follow-up to this question I would like to clarify whether the tangent bundle on a sphere in $\mathbb R^3$ spans $\mathbb R^3$ to make sure I get the concept. The tangent bundle is the set of the tangent planes at every single point on the…
Antoni Parellada
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Trying to understand the first fundamental form

I have read similar questions with regards to what the first fundamental form is. I couldn't find my answers due to them assuming extra knowledge and/or using a different book which presents the topics differently. I am studying differential…
user462561
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Riemann manifold

Let's suppose we have $N$ a compact Riemann manifold and a smooth function f on N. Prove that $\nabla f= 0$ at 2 or more points. I am not very sure that this question is correct because I don't see how the fact that N is Riemannian fits.
user53970
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Hypersurface orientable if it admits a smooth normal vector field

Let $X$ a codimension 1 smooth submanifold of the n-dimensional smooth manifold $Y$. Assume $Y$ is oriented. We want to show that $X$ is orientable if and only if it admits a global smooth normal vector field (in Y). How can we prove this? I have no…
Bernard
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Periodic parametric curve on cylinder

Given a cylinder surface $S=\{(x,y,z):x^2+2y^2=C\}$. Let $\gamma(t)=(x(t),y(t),z(t))$ satisfy $\gamma'(t)=(2y(t)(z(t)-1),-x(t)(z(t)-1),x(t)y(t))$. Could we guarante that $\gamma$ always on $S$ and periodic if $\gamma(0)$ on $S$?
coba
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Yang-Mills equations

I would like that someone explain to me the Yang-Mills equations as defined in some books: $$ \begin{cases} d_D F = 0 \\ *d_D *F = J \end{cases} $$ What is $ d_D $? What is $ F $? What are $ *d $ and $ *F $? What is $ J $? Can we represent those…
Bryan
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Embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$ (2)

Related question: embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$ I want to show $F[x,y,z]=(x^2−y^2,xy,xz,yz)$ is an immersion. I know how to do it in local charts, for example by computing the jacobian of the composite map $(x,y) \mapsto…
Gobi
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Doubly Ruled Surfaces

I get stuck in the following exercice: Let $S$ be a doubly ruled surfaces by orthogonal lines. Use the Gauss equations to prove that $K\equiv 0$; Conclude that $S$ is a plane. I tried to use a parametrization $X(u,v)=\alpha(u)+v\beta(u)$, usinge…
Jön
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Product of Smooth Covering Maps a Smooth Covering Map? ( J.Lee, 2-12)

I am trying to review my differential Geometry. In J.Lee's Smooth Manifolds, there is an exercise in which one has to show that the product of smooth covering maps is a smooth covering map. A smooth covering map is a smooth cover $\pi: M'…
user7679
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Pre-image of a submanifold by a submersion

Let $f: M \to N$ a submersion between two manifolds, and let $S\subset N$ a subset of N. Proof that $S$ is a regular submanifold of $N$ if and only if $f^{-1}(S)$ is a regular submanifold of M. So far I have half of the problem. Using the…
alexp9
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