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)

1
vote
0answers
8 views

Is my proof that the sheaf of smooth functions is soft correct?

As mentioned in the title, I proved the following statement and need help from someone who checks it for correctness: Let $X$ be a paracompact smooth manifold. Then $U \mapsto C^\infty(U)$ where $U$ ...
1
vote
0answers
24 views

What is the step in this proof “because $\omega$ is closed”?

I am working through this proof of the Poincare lemma here but I don't understand one step. First, there is the following equation $$ {\partial \over \partial x^j} f(x) = \int_0^1 \left (t ...
1
vote
1answer
16 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
0
votes
0answers
57 views

Show $f:\mathbb{R}^n \to \mathbb{R}^m$, $n>m$ can't be 1-1

Problem 2-37 on p. 39 of Spivak's Calculus on Manifolds asks Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuously differentiable function. Show that $f$ is not 1-1. (Hint: If, for example, ...
2
votes
1answer
29 views

When is a curve parametrizable?

Is there a way in general to tell whether a given curve is parametrizable?
0
votes
0answers
16 views

Tangential derivative vs covariant derivative

My question is basically the same as this, but the answer in that page was not clear to me. Let me restate the question here: let $\Omega\subset\mathbb{R}^3$ be a domain with boundary $\Gamma$, and ...
-1
votes
2answers
34 views

Notation of coordinate representation in Lee

In Lee's Introduction to Smooth Manifolds he writes $$ \omega = \omega_i dx^i$$ where $\omega$ is a differential form. See for example page 293. What does $\omega_i dx^i$ stand for? According ...
1
vote
0answers
38 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
1
vote
0answers
20 views

Name of isometric invariant in Gauss-Bonnet

Does the tangential rotation term $ \int k_g ds $ of Gauss-Bonnet theorem ( for continuous or discontinuous lines on a surface) have a name or symbol in differential geometry ? The second term $ ...
2
votes
1answer
38 views

Can I argue like this to prove that the determinant is positive?

Let $X$ be a smooth $n$-manifold with an oriented atlas $\mathcal U = (U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$. Let $g$ be a Riemannian metric on $X$. Let $g_{ij} = g\left ( {\partial \over ...
2
votes
1answer
85 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
0
votes
1answer
23 views

Flows: Escape Lemma

Denote compact sets by: $\mathcal{C}$ Given a smooth manifold. Consider the maximal flow of a smooth vector field: ...
0
votes
1answer
29 views

Simple Differential Geometry/Analysis question: Prove that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous

In Differential geometry of curves and surfaces by Manfredo do Carmo, page 459, it says the following: Observe that $f:\mathbb{R^2}\to\mathbb{R}$ is continuous, where $f(x,y) = \frac{x^2}{a^2} - ...
0
votes
0answers
22 views

Parametrization of surfaces gauss

What is pde relating $f$ and $g$ if the Gauss curvature is $+1$ and $-1$ respectively: $$ \begin{align*} (x,y,z) &= ( u \cos(v), u \sin(v), f(u,v) ) \\ (x,y,z) &= ( u \cos(v), u \sin(v), ...
3
votes
1answer
33 views

What is the anticommutator of the interior product and codifferential (adjoint of exterior derivative)?

What is $\eta=i_X \delta + \delta i_X$ acting on differential forms? Here $\delta$ is the usual Hodge adjoint of the exterior derivative and $i_X$ is contraction of a form with the vector field $X$. ...
2
votes
1answer
19 views

Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
2
votes
1answer
61 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
0
votes
1answer
22 views

Paramatrizes curve with constant speed

Show that if $\alpha : I \rightarrow \Re^{n+1}$ is a parametrised curve with constant speed then $\alpha(t) \perp \frac{d}{dt} \alpha(t)$ for all $t\epsilon I$.
2
votes
1answer
45 views

First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
0
votes
2answers
33 views

Why are derivations useful for defining tangent vectors?

On page 54 in his book Introduction to Smooth Manifolds, John Lee says the following: A linear map $v: C^\infty (M) \rightarrow \Bbb{R}$ is called a derivation at p if it satisfies ...
3
votes
1answer
39 views

Map between Tangent Manifolds Well-Defined?

Let $f: \mathcal{M} \to \mathcal{N}$ be a $\mathscr{C}^{r+1}$ map. We define a map $\mathscr{T}f: \mathscr{T}\mathcal{M} \to \mathscr{T}\mathcal{N}$ as follows: A local representation of the map ...
0
votes
1answer
27 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...
1
vote
2answers
90 views

If the curvature of a curve in a plane is a constant, is the curve contained in a circle?

If the curvature of a curve in a plane is a constant, is the curve contained in a circle?(Suppose curvature is positive.) one of my homework problems needs to use this, but I am not sure whether this ...
1
vote
0answers
22 views

Recover a surface's equation from its curvature

Can the equation of a surface in Euclidean 3 space be recovered from the equation of its Gaussian curvature?
1
vote
1answer
50 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
0
votes
1answer
30 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
1
vote
1answer
52 views

What class would this be covered in?

Would the material in this section of a wikipedia article be covered in a standard course on Differential Geometry, or should I look elsewhere to learn those sorts of things? Specifically, topics like ...
1
vote
1answer
47 views

Gluing submanifolds along their common boundary

Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that $N_1\cap N_2=\partial N_1=\partial N_2$, for each $x\in N_1\cap N_2$, $T_x N_1=T_x N_2$, and for $x\in ...
3
votes
1answer
41 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
0
votes
0answers
44 views

Exterior Algebra: Characterization [on hold]

Just a short question: Given a vector space. Why is the exterior algebra characterized as: The largest anticommutative integer-graded algebra with identity linearly embedding the vector ...
2
votes
2answers
46 views

A doubly ruled surface which is not a plane must be quadratic

I want to show that a doubly ruled surface which is not plane must be quadratic. Any help will be appreciated.
1
vote
1answer
61 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
0
votes
2answers
51 views

Prove that $\frac{dN}{ds}=-\kappa T$

Prove that $\frac{dN}{ds}=-\kappa T$, where $N$ is the oriented normal, and $T$ is the unit tangent vector, and $s$ is arc-length parameter. Here's what I've got so far from my note and I don't ...
3
votes
3answers
44 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
1
vote
2answers
43 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
6
votes
1answer
69 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
3
votes
0answers
40 views

When is a linear map of 1-forms a pullback?

Every diffeomorphism $\phi: M\to N$ between two-dimensional compact oriented Riemannian manifolds induces a linear map on one-forms $L:\Omega^1(M)\to\Omega^1(N)$ given by the pullback of $\phi^{-1}$. ...
2
votes
1answer
29 views

What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

differential geometer in training here. With regards to my background, I learned differential and Riemannian geometry from O'Neill and Lee's series. I'm working on my algebra background (which is ...
0
votes
3answers
32 views

Is saying each pt of a topo. manifold has nbhd homeomorphic to R^n the same thing as saying there is a local coordinate system at each point?

Is saying each point of a topological manifold has a neighborhood homeomorphic to $\Bbb{R}^{n}$ the same thing as saying there is a local coordinate system at each point? I'm not really sure what ...
3
votes
2answers
47 views

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R^3}$ $x^2+y^2+z^2=1$ and $x^2+y^2=x$.

Find a parametrization of the intersection curve between two surfaces in $\mathbb{R}^3$ $$x^2+y^2+z^2=1$$ and $$x^2+y^2=x.$$ I know that $x^2+y^2+z^2=1$ is a sphere and that $x^2+y^2=x$ is a circular ...
0
votes
1answer
62 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
0
votes
1answer
63 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
1
vote
0answers
24 views

Zero Gauss curvature surfaces spanning a loop

EDIT 1: Spanning across a given arbitrary closed boundary/loop a surface can be defined with zero mean curvature H in 3-space as minimal surfaces. Likewise can a surface be defined with zero Gauss ...
4
votes
1answer
71 views
+50

Hessian of a function on Riemannian manifolds

Let $(M,g,\nabla)$ be a Riemannian manifold with metric $g$ and Riemannian connection $\nabla$. The hessian of a function $f:M\to R$ is defined by: $$H^f(X,Y)=g(\nabla_X\ \ \operatorname{grad} ...
0
votes
1answer
14 views

vertical/horizontal asymptotes - general understanding

Do vertical asymptote only exists in fractions? My taught was yes. Can the curve cross a vertical asymptote? My taught was no. Can the curve cross a horizontal asymptote? my taught was yes. Thanks
1
vote
2answers
27 views

Embed curves in the plane

The strongest version of Whitney's embedding theorem says that every smooth real $n$-dimensional manifold $M^n$ (Hausdorff and second-countable) can be embedded in $\mathbb{R}^{2n}$. This should mean ...
0
votes
1answer
13 views

Must a Developable Surface be Tangent Developable or a Generalised Cone/Cylinder?

I've commonly seen that tangent developable surfaces, Generalised cones and generalised cylinders are developable surfaces. (see http://en.wikipedia.org/wiki/Developable_surface) But are these the ...
2
votes
0answers
55 views

Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet) Please feel free to recommend ...
4
votes
2answers
95 views

How to evaluate this integral: $\oint dx$?

I am trying to understand differential forms. Now I tried to evaluate $$ \oint_{S^1}dx$$ I should get anything non-zero but I don't know how to do it (even though I know the result). If $S^1$ in ...
0
votes
0answers
21 views

Use Frenet Frame and Pythagorean Theorem.

Suppose we have a curve $c(t)$ where $t$ goes from $a$ to $b$. $c$ has positive curvature and a frenet frame(TNB). Choose $\rho > 0$ and small. and define: $f(t) = c(t) +\rho B(t)$ $g(t) = c(t) ...