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
1answer
77 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
2
votes
1answer
145 views

Is this an abuse of notation?

Here is a proof says that the differential of Gauss map is self-adjoint. But I seems there is an abuse of notation at (1) in it. Since $dN_p$ is linear, it suffices to verify that $\langle ...
0
votes
1answer
79 views

Curvature of plane curve

This is a motivation for the definition of the curvature of a plane curve: Suppose then that $\gamma$ is a unit-speed curve in $\mathbb{R}^2$. As the parameter $t$ of $\gamma$ changes to $t+\Delta ...
-1
votes
1answer
27 views

How to find the radius in a hypersphere given an arc

Does anybody know how to calculate the radius in a hypersphere if the arc length is known and the curvature of the arc?
0
votes
1answer
49 views

Geometric changes

I want to find the coordinate of point on filament yarn after receiving some force or heat . please help me !
0
votes
1answer
134 views

Why does a singular point create a cusp or a node on the trace?

What the geometrical meaning for a singular point of a parametric curve? i.e Suppose $\alpha$(t) = (x(t), y(t)), then $\alpha '$ (t) = (x'(t), y'(t)). A singular point at t$_0$ is when $\alpha ...
1
vote
1answer
134 views

What is the one form given its value for a vector field?

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we ...
10
votes
0answers
314 views

Big geometry grad schools - for an average applicant [closed]

What are some schools that have a lot of geometry going on, but that might accept some middle-of-the-range applicants? Let me add some context... I left grad school (UC Davis) with an MS in 2012 ...
2
votes
1answer
132 views

finite length of a spiral

consider a "spiral" $\alpha(t)=r(t)\left(\cos(t),\sin(t)\right)$, where $r$ is $\mathcal{C}^1$ and $0\le r(t) \le 1$ for all $0 \le t$ Show that if $\alpha$ has finite length on $ [0,\infty)$ and ...
2
votes
1answer
70 views

There exists a constant arc length parametrization

I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. ...
1
vote
1answer
64 views

Properties of Curvature

I'm having some trouble understanding how to approach this problem. Any help is appreciated. Suppose $\alpha$ is an archlength- parameterized space curve with the property that ...
2
votes
0answers
125 views

Moebius strip as a fibre bundle

I've alrealdy asked this question, but now I have more clear ideas, so I'm going to ask again and see if I'll understand a bit more. It's about the trivialisation of the Moebius strip as a bundle on ...
1
vote
3answers
461 views

What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?

What is the definition of $R_{ijkl}$ in terms of metrics on a manifold? I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?
1
vote
0answers
57 views

Vector fields on homogeneous space $G/H$

I am trying to understand why the vector fields on $G/H$ are maps $X:G\rightarrow \frak{g}/\frak{h}$ satisfying $X(rh)=Ad^{-1}(h)X(r),\,\, h\in H.$ Any hint would be greatly appreciated!
4
votes
1answer
186 views

The Definition of the Second Fundamental Form

Let $r:M\rightarrow{\mathbb{R}^{n+1}}$ be an isometric immersion and $M$ is an $n$-dimensional Riemannian Manifold. That is to say, $M$ is the hypersurface in $\mathbb{{R}^{n+1}}$. Then we can ...
0
votes
0answers
64 views

Finding the components of the Riemannian tensor given the components of a metric.

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
0
votes
0answers
103 views

Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
2
votes
1answer
405 views

Tangent space of Grassmannian $Gr_k (\mathbb{R}^n)$

I am trying to show that the tangent space of the Grassmannian $Gr_k (\mathbb{R}^n)$ at $L,$ is naturally/canonically isomorphic to $Hom(L,\mathbb{R}^n/L).$ However, I cannot see even intuitively what ...
1
vote
2answers
86 views

Four circles & a square in a circle

Radius of the big triangle is $2$. ABCD is a square. What is the difference between $T_{1}$ and $(M_{1}+M_{2})$. I have solved it already though I don't know if my answer is right or wrong. My ...
3
votes
2answers
120 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
3
votes
1answer
110 views

Flat Points in Irreducible Algebraic Varieties

I am trying to understand the paper "Algebraic Methods in Discrete Analogs of the Kakeya Problem" by L. Guth and N. H. Katz. This paper contains the following lemma: Let $S$ be the set of points in ...
2
votes
2answers
98 views

What does “flat hypersurface” mean?

If $S$ is a flat hypersurface with boundary in $\mathbb{R}^n$, what does it mean? Is it just a simple open domain (found in most PDE contexts)?
0
votes
1answer
101 views

Diffeomorphism of closure of open sets

Let $F:\overline{X} \to \overline{Y}$ be a map between the closure of two open Lipschitz domains $X$ and $Y$ in $\mathbb{R}^n$ (with boundaries). $F$ is such that it maps $X$ to $Y$ and it maps ...
2
votes
0answers
110 views

Formula for integral over hypersurface??

Can someone give me a formula for an (Lebesgue) integral of a function $f:M \to \mathbb{R}$ where $M$ is a bounded $C^k$-hypersurface of dimension $(n-1)$ in $\mathbb{R}^n$? I have tried the ...
2
votes
1answer
140 views

Tangent space of a flag manifold?

I am studying differential geometry and now I am trying to find the tangent space to a flag manifold $F(a_1,a_2,...,a_k, \mathbb{R^n}).$ Any hint would be greatly appreciated!
0
votes
1answer
87 views

definition of a map from $CP^1$

I think this is a very easy question, but I've got problems understanding how the function in the second exercise of this pdf (that I found online on google and I wanted to try in order to improve my ...
7
votes
2answers
256 views

When does homeomorphism imply diffeomorphism?

In $R^n$, suppose $U$ and $V$ and two homeomorphic open sets. Then, is $U$ diffeomorphic to $V$? If not, can we impose stronger conditions such that this true?
5
votes
1answer
241 views

Christoffel Symbols Equality Solution?

They changed the exercise, so I tried to solve it again: I have to prove the following: Let $\Omega \subseteq \mathbb{R}^d$ be open and $g$ a metric field on $\Omega$. For every $\phi \in ...
3
votes
1answer
89 views

D'Alembertian $\Box$

This question has to do with the D'Alembertian operator on a general manifold with a metric $g_{\mu\nu}$. I understand that the definition of the D'Alembertian is $$\Box \phi\equiv ...
0
votes
2answers
79 views

How to prove $H^1(M) \subset H^s(M)$ is a continuous embedding for manifold $M$?

Let $M$ be a $C^k$ manifold for some integer $k$. How does one show that $$H^1(M) \subset H^s(M)$$ is continuous, where $s \in (0,1)$? I was planning to pull back the norms onto a subset $D_i$ of ...
2
votes
1answer
188 views

Why is the space of all connection on a vector bundle an affine space?

I think this result is very well known, but I don't understand its proof. Let E a vector bundle over a manifold M, and $\Omega^i(E):=\Gamma(\Lambda^iT^*M\otimes E)$ the space of E-valued differential ...
0
votes
1answer
488 views

Hodge double star operator

I want to prove that $**\omega=\left(-1\right)^{k\left(n-k\right)}\omega$, where $*$ is the Hodge star operator acting on differential $k$-forms $\omega$ on $\mathbb{R}^n$. Where can I find the proof ...
1
vote
0answers
166 views

Some Advice on My Undergraduate Paper

My teacher wants me to read something about "Differential Geometry in $R^3$" and choose a topic as a paper. Now I have finished these books. And I am interested in some topics below: $(1)$ ...
-3
votes
1answer
391 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
2
votes
1answer
108 views

Metric on tangent vectors to tangent space

Let $M$ be a Riemannian manifold and $p$ be a point of $M$. Let $v$, $v'$ be tangent vectors to $M$ at $p$. Of course we have $\langle v,v'\rangle_p$ defined. Let $u$, $w$ be tangent vectors to ...
2
votes
1answer
150 views

Showing that a subset of the real projective plane is a smooth manifold under given condition

I'm trying to solve exercise 9.7 in Tu's introduction to manifolds: Let $F(x_{0},x_{1},x_{2})$ be a homogeneous polynomial of degree $k$. Consider the homogeneous coordinates ...
2
votes
1answer
80 views

A connection over a 1-dim manifold is flat

Let $M$ be a 1-dimensional manifold and let $E$ be its vector bundle. I want to show that every connection $D$ on this vector bundle is flat. A connection $D$ is flat means that we have $$D_v D_w ...
1
vote
1answer
49 views

Lipschitz map between hypersurfaces/manifolds

if $A$ and $B$ are compact hypersurfaces or manifolds and $F:A \to B$ is a $C^1$ diffeomorphism, does it follow that $F$ is Lipschitz? I am think of the case where these hypersurfaces are boundaries ...
1
vote
1answer
110 views

Flow of a vector field: how existence of a flow line implies existence of flow.

I am unable to see why there exists $U$ such that $\phi_t(x)$ exists for all $t\in[0,T]$. Can you please help me to understand the argument above. Thanks.
1
vote
2answers
80 views

Is not the surjective map $\pi$ associated with a vector bundle infact a bijection?

I am reading John M Lee's Riemannian Manifolds : An Introduction to Curvature, which is very well written. On page 16 : "Vector bundles are defined", quoting A (smooth) $k$-dimensional vector ...
20
votes
3answers
292 views

Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
1
vote
1answer
141 views

Submanifold is complete

If $M$ is a complete manifold and $N\subset M$ is a closed, embedded submanifold with the induced Riemannian metric, show that $N$ is complete. I really don't know where to start. This is not ...
3
votes
1answer
84 views

Differentiable parameterization of a curve $\Gamma$

If $\alpha:I\longrightarrow \Gamma$ and $\beta:J\longrightarrow \Gamma$ are two bijective differentiable regular parameterizations of the curve $\Gamma\subset\mathbb{R}^2$ (not necessarily of class ...
0
votes
1answer
102 views

About integration on manifold and partition of unity (and finiteness of open covers)

Please see the definition below of integration over a boundary of a Lipschitz domain. My question is, the summation in (C.36) for example is over $n$. But when is this a finite sum? If ...
1
vote
2answers
75 views

invariant inner product on eigenspace

I have several questions about the following corollary: "Let G/H be a riemannian homogeneous space where G is a compact Lie group. Let $E_{\lambda}=\lbrace f\in C^{\infty}(G/H) : -\Delta f= \lambda ...
2
votes
2answers
245 views

Principal bundle automorphism generating global gauge transformations

Consider a principal $G$-bundle $P$ with connection form $\omega$. An automorphism $f$ of $P$ is by definition a (smooth) $G$-equivariant map: $f(p \cdot g) =f(p) \cdot g$ for all $p\in P$ and $g\in ...
2
votes
2answers
263 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...
0
votes
1answer
40 views

$C^{0}$-norm of a map

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ? Greetings, Daniel
1
vote
1answer
135 views

Exterior product of n copies of 2-form

I have a problem with calculating exterior product of differential forms. Here is the problem: Let $\omega$ be a 2-form in $\mathbb{R}^{2n}$ given by $\omega=dx_{1}\wedge dx_{2}+dx_{3}\wedge ...
6
votes
2answers
396 views

Is there a field of 'real analytic geometry'?

I am wondering whether there is a field of 'real analytic geometry', and if not, why not? There are branches of geometry corresponding to increasingly large sets of functions: polynomial (algebraic ...