Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

learn more… | top users | synonyms (1)

0
votes
0answers
16 views

Dual coordinates

If we have $R^{4}$ with basis $\{e_{1},e_{2},e_{3}=Je_{1},e_{4}=Je_{2}\}$. If we choose a surface $S=\{e_{1},e_{2}\}$ in $R^{4}$ and $x^{i}$ the coordinates dual to basis $e_{i}$. In order that S be ...
0
votes
0answers
28 views

Express the change of coordinate matrix in terms of partial derivative of the transition map $\phi$ Where $\tilde{\sigma} =\sigma \circ \phi$

Let $\sigma :U \to W\cap S$ and $\tilde{\sigma}: \tilde{U} \to \tilde W \cap S$ Be two surface patches around $p\in W\cap \tilde W$ $T_p(S) $ be tangent plane of S at p. $T_p(S)=span\{\sigma_u, ...
0
votes
0answers
51 views

chain rule for differential forms?

Let $f:U \to V$ be (smooth) map, where $U,V \subset \Bbb{R}^n$ are open subsets. Let $X(t)\in U$, then I saw an equation like $$\frac d {dt} f(X(t)) = df(X(t))X'(t)$$ But I'm not understanding what ...
0
votes
0answers
152 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
0
votes
0answers
58 views

find the estimated coordinate from a transformed system

Hello let me explain the situation We have the cartesian coordinate system in A. And then, it was transformed to become B. B as you can see is a transformed version of A. The red points signifies the ...
0
votes
0answers
65 views

Confusion on the meaning of “the opposite sides” in below problem.

Below is a problem which states a fact about "Tchebyshef net". I don't understand meaning of bolded part. Please help me. The coordinate curves of a parametrization $x(u, v)$ constitute a ...
0
votes
0answers
65 views

Inverse of a pythagorean hodograph curve.

I'm trying to solve this question about pythagorean hodograph curves: If $\alpha$ is a Pythagorean hodograph curve, and $\alpha(t)\neq(0,0)\;\forall t$, then the curve $\frac{1}{\alpha(t)}$ is also a ...
0
votes
0answers
30 views

arbitrary patch in terms of orthogonal patch

Let $\mathcal{M}$ be a $2$ dimensional differentiable manifold (a geometric surface). Let $\mathbf{y}$ be an arbitrary patch in $\mathcal{M}$. How can one prove that $y$ can be expressed as ...
0
votes
0answers
19 views

Existence of an integratiag factor (in the proof of isothermal parameters in analytic case)

Suppose $f(x,y),g(x,y)$ are analytic functions, does there exist a function $\lambda(x,y)$, such that $\lambda(fdx+gdy)=dh$ for some $h(x,y)$?(locally)
0
votes
0answers
52 views

area of a flat torus of revolution

Let $R>r>0$. We get a torus $T_0$ from the map $F(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u)$. Now define a new metric tensor on $T_0$ by $(x_u,x_u)=(x_v,x_v)=1, (x_u, x_v)=0$. ...
0
votes
0answers
49 views

Stuck on proving a surface is a cylinder or a plane

A surface with first and second fundamental forms $\mathrm{I}$ and $\mathrm{II}$ whose coefficients(i.e. $E,F,G,L,M,N$ with $\mathrm{I}=Edudu+2Fdudv+Gdvdv,\mathrm{II}=Ldudu+2Mdudv+Ndvdv$) are all ...
0
votes
0answers
111 views

Vector field and triangulation

I'm writing a paper on Poincaré-Hopf theorem about vector field indices and Euler Characteristic over topological compact surfaces and I got struck finding details on the last part of the proof. Let ...
0
votes
0answers
52 views

Curve on $S^2$ (2-sphere)

Let $\gamma$ be a curve on the 2-sphere $S^2$. Then: i) $k_n = |\gamma''|$ is constant. ii) $k_g=\langle t'(s),n(s)\rangle$ is constant. iii) $\gamma$ is a geodetic line, i.e. $k_g\equiv 0$ if ...
0
votes
0answers
42 views

Angle between unit vectors

I have a differential geometry problem... Given two unit vectors $X^r$ and $X^r+dX^r$ of the same curve, show that the angle between them is: $\theta^2=g_{mn}dX^mdX^n$. I understood how to do it with ...
0
votes
0answers
35 views

Definition of standard connection on $\mathbb{R}^n$ vs acceleration vector

I am currently looking at a set of lecture notes where the standard covariant derivative $D_XY$ in $\mathbb{R}^n$ is defined to motivate the more general concept of a connection. For two vector ...
0
votes
0answers
29 views

Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
0
votes
0answers
37 views

Differentiation along a curve on a manifold (Re: Schutz's intro to GR)

I am trying to show (Schutz chpt. 6 prob 13) that if two vector fields $\vec{A}$ and $\vec{B}$ are parallel transported along a curve $\gamma:\mathbb{R}\to M$ with real parameter $\lambda$ ($M$ a ...
0
votes
0answers
18 views

Continuous groups of Transformations [Reference request]

I am considering reading the book : 'Continuous Groups of Transformations' by Luther Pfahler Eisenhart. It seems to have a very interesting table of contents. However this is quite old and I am ...
0
votes
0answers
149 views

Tangent surfaces locally isometry on the plane

Let $\alpha : I \rightarrow \mathbb{R^3}$ be regular parametrized curve with the curvature $k(t) \neq 0$, $t\in I$. Let $$ \mathbb{X}(t,v) = \alpha(t) + v \alpha'(t) $$ be tangent surface. Prove ...
0
votes
0answers
179 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
0
votes
0answers
42 views

A Survey on Rota's Conjecture

I am looking for a survey on Rota's Conjecture (1970). I am more interested in geometric aspects of the conjecture and any geometric content related to the conjecture. Any reference would be helpful. ...
0
votes
0answers
46 views

kernel of a smooth map

Let $M,N$ be smooth manifold , and let $f:M\rightarrow N$ be a smooth map, such that $y\in N$ is a regular value of it. Denote $M'=f^{-1}(y)$. Lemma: $ker(df_{x})=TM'_{x}$ for every $x\in M'$, ...
0
votes
0answers
152 views

Preimage of a regular value is manifold , for smooth map between smooth manifolds

I was reading the following Lemma: Let $M,N$ be manifold of $m\geq n$ dimensions respectively , and let $f:M\rightarrow N$ be smooth, then for every regular value $y\in M$ , the Preimage $f^{-1}(y)$ ...
0
votes
0answers
48 views

principal curves of a flat surface.

Let M be a surface in $R^3$ with principal curvatures $k_1=0, k_2\neq 0$. $k_1$ is always zero and $k_2$ is never zero. Suppose that $E_1, E_2$ are corresponding principal unit vector fields and ...
0
votes
0answers
26 views

conformally flat 3-manifolds

Does there exist closed three manifolds that don't admit a metric that is locally conformally flat? Recall that a three manifold is locally conformally flat if the exterior derivative of the Schouten ...
0
votes
0answers
31 views

Tangent space on a differentiable function

I am wondering about a definition that I found in my notes: Let $f$ be a differentiable function at a point $x_0$ and $Df(x_0)$ denote its derivative, then $\text{Graph(Df(x_0))}:=\{(h,Df(x_0)h)\}$ ...
0
votes
0answers
71 views

Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
0
votes
0answers
32 views

An abstract definition of the cotangent space to a smooth manifold.

I need a book introducing the cotangent space to smooth manifolds in the most abstract way. So $T^\ast_p M$ by this point of view should be the quotient ring $I/I^2$ where $I=\{[f]\in ...
0
votes
0answers
57 views

Normal curvature of a circle in a plane

I have the circle $\gamma(t) = (\cos t, \sin t, 0)$ in the plane $z=0$. Now I understand that normal curvature is related to the second fundamental form, and an expression for it is ...
0
votes
0answers
44 views

lie group homomorphism and compactness

Let $f\colon M\rightarrow N$ be a surjective homomorphism of lie groups. Assume the follwoing: f has discrete kernel $ker(f)=\{e,-e\}$ a path from $e$ to $-e$ exisits $N$ is connected Claim: $M$ ...
0
votes
0answers
26 views

Manifolds : Show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique.

So I want to show that a maximal $C^r$-atlas containing a $C^r$-atlas $A$ is unique. I believe I have shown this, but in my proof I require the fact that the charts of $A$ cover our manifold in ...
0
votes
0answers
43 views

Volume and Ricci curvature

If the Ricci curvature is zero on a manifold, does that mean that, if I choose a sphere and let each point contained in that sphere move along geodesics that are all initially parallel, the volume of ...
0
votes
0answers
170 views

total Gaussian curvature of a surface

I have to evaluate a total Gaussian curvature of the surface in $\mathbb{R}^3$ : $x^2 + y^2 - z^2 = 1$ I tried to solve the problem by using Gauss-Bonnet Theorem or direct computation. When I used ...
0
votes
0answers
16 views

Chain of transformations -> continuous

Transformation $A_t$ rotates point $p(t)$ for angle $d\phi(t)$ around the axis $n(t)$ anchored at point $r(t)$ and finally displaces it for $r'(t) dt$. Point is now $p(t + dt)$. More specifically in ...
0
votes
0answers
62 views

French translation, and what is the curvature of a metric?

I have a french paper to read. There is the notion of une collection des courbures des métriques $g_t$. Now I would guess that this refers to a collection of curvatures of metrics $g_t$, however ...
0
votes
0answers
38 views

Vector bundle addition

Let $\pi:E \to M$ a smooth vector bundle and let $E \times _M E= \{(x,y) \in E \times E| \pi(x)=\pi(y) \}$. I want to show the addition map $E \times_M E \to M, (x,y) \to x+y$ is smooth. The ...
0
votes
0answers
24 views

Examples of totally real minimal submanifolds

If we have $R^{4}$ with basis $\{e_{1},e_{2},e_{3}=Je_{1},e_{4}=Je_{2}\}$, then we know that $\{e_{1},e_{2}\}$ is totally real minimal submanifold of $R^{4}$. Is there a nontrivial example of totally ...
0
votes
0answers
111 views

Liouville form on the cotangent bundle

a) Let $(U,\phi) = (U,x^1,\dots,x^n)$ be a chart on a manifold $M$, and let $(\pi^{-1}U,\tilde {\phi}) =(\pi^{-1}U,\tilde {x^1},\dots,\tilde {x^n},c_1,\dots,c_n)$ be the induced chart on the ...
0
votes
0answers
40 views

A curve is contained in a hyperplane if and only if $\kappa_{n-1} = 0$

I'm trying to prove that a curve $\gamma$ in $\mathbb{R}^n$ is contained in a hyperplane if and only if $\kappa_{n-1} = 0$, where $$\kappa_{i} = \frac{\langle e_{i}',e_{i+1}\rangle}{\| \gamma' \|}$$ ...
0
votes
0answers
60 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
0
votes
0answers
65 views

how to determine the curve

I have to determined the curve which passes through the point $(1/2, \sqrt3 /2)$ and cuts to each member of the family of circles $x^2+y^2=a^2$ forming a angle of $60º$ My idea is to create a ...
0
votes
0answers
50 views

question about regular mapping in elementary differential geometry by Oneill

I am looking at Oneill elementary differential geometry section 4.2 Patch Computations. In example 2.4, parametrization of a surface of revolution, it says Suppose that $M$ is obtained by revolving ...
0
votes
0answers
31 views

group of automorphism of an $SU(2)$-bundle

Let $H$ be the Hopf bundle over $S^2$, consider the vector bundle $H^k \bigoplus H^{-k}$ over $S^2$ and extend it to $S^2 \times R^+$, this gives an $SU(2)$-bundle $E_k$. The claim is that the group ...
0
votes
0answers
82 views

Dupin's indicatrix and asymptotic direction of a surface?

Asymptotic direction at a point $p$ of a surface $S$ is defined to be the direction of $T_{p}(S)$ for which the normal curvature is zero. And Dupin's indicatrix at a point $p$ of surface $S$ is ...
0
votes
0answers
53 views

Sections on the Tautological Line bundle $E(\gamma_n)$..

I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb ...
0
votes
0answers
25 views

Why is positivity of $g$ required for $(x,v) \mapsto (x,g(x)v)$ to be smooth?

An exercise in Guillemin and Pollock (1.8.2) assumes that $g$ is a smooth, everywhere positive function on a manifold $X$. The book assumes all manifolds are embedding into some ambient Euclidean ...
0
votes
0answers
47 views

How to compute the scalar curvature of this submanifolds?

Hello : I would like to know how to compute the scalar curvature of the following submanfolds $ H = \{ ( t , x,y,z) \in \mathbb{R}^4 \ : \ t = ax^2 + by^2 + cz^2 \} $ of $ \mathbb{R}^4 $ using the ...
0
votes
0answers
75 views

Foliations transverse to a fiber bundle

In the book Geometric Theory of Foliations by Camacho and Neto a foliation $\mathcal{F}$ is called transversal to a fiber bundle $\pi: E \rightarrow B$ with fiber F if a)For every p in E the leaf ...
0
votes
0answers
63 views

Show that the inverse function is not continous.

I am working with the following exercise: Let $U = \{ (u,v) \in \mathbb{R}^{2} | -\pi < u < \pi, 0 < v < 1\}$, define $X: U \rightarrow \mathbb{R}^{3}$ by $X(u,v) = (\sin u, \sin 2u, v)$ ...
0
votes
0answers
102 views

How to prove the following function is a diffeomorphism.

I am trying to prove that n-closed unit ball is Manifold with boundary. I constructed a function as follows. (Using Stereographic projection idea) $N$ is the north pole i.e, $ (0, \dots, 1)$ $X$ be ...