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
11 views

a differential geometry question, is $k_\alpha(s) \ge |k_\beta(\alpha(s))|{d\sigma\over ds}$ true or wrong?

let $ \alpha(s)$(s is the arc-length of $\alpha$) be a closed curve in three dimensional space, let $\beta$ be orthogonal projection of $\alpha(s)$ in xy plane. if so, we have ...
4
votes
1answer
33 views

Definition of covering (deck) transformation for smooth manifolds: Are they diffeomorphisms?

In John Lee's book Riemannian Manifolds, a covering transformation (or deck transformation) of a smooth covering map $\pi:\tilde{M}\to M$ (of connected smooth manifolds) is defined to be a smooth map ...
-1
votes
0answers
32 views

Research areas lying at the confluence of Analysis and Geometry

I wanted to get expert opinion on what are the areas of active research lying at the confluence of Analysis and Geometry. Two areas that I have heard about are : (1)Geometric Analysis and ...
1
vote
2answers
36 views

Symbol $\Gamma$ when talking about vector fields.

I noticed several times online that people tend to use the symbol $\Gamma(M,TM)$ when talking about the space of smooth vector fields on smooth manifolds. I find this totally confusing, as in ...
0
votes
0answers
22 views

Scaling of minimal surfaces

After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact. Is the statement ...
2
votes
1answer
28 views

Definition of submanifolds by regular values

Let $f: M \rightarrow N$ and $q \in N$ be a regular value, then $f^{-1}(q)$ is a submanifold of $M$. Now assume that $q \in N$ is not a regular value, but you pick $K:=f^{-1}(q) \cap \{p \in M; ...
2
votes
1answer
43 views

Homeomorphism between $S^2$ and $CP^1$ via uniqueness of quotient

I am trying to show that $S^2$ and $\mathbb{C}P^1$ are homeomorphic making use of the following result - see e.g. Jack Lee Introduction to topological manifolds. Let $Y \xrightarrow{\pi_1} X_1 $, $Y ...
0
votes
0answers
18 views

Question on application of the Morse lemma

Consider the following example of an application of the Morse lemma (to plane function): Consider two unit speed pieces of curve (perhaps part of the same closed curve) parametrized by ...
1
vote
0answers
14 views

Significance of sasaki manifold, kenmotsu manifold, einstein manifold and different curvature tensors

I have studied different sasaki manifold, kenmotsu manifold and einstein manifold and also different curvature tensors during my masters and also worked on the above domains. However working on ...
2
votes
0answers
19 views

equivalent definitions of tensorfields

I have been confused about something for quite some time now and I would really much appreciate to get a clear explanation of the following. There are two equivalent definitions of how to define ...
-1
votes
0answers
25 views

The equation of hypersufaces in $R^{4}$ [on hold]

Can this be an equation of hypersurface in $R^{4}$: $x_{1}+x_{2}=c$, $c$=const.
3
votes
1answer
40 views

Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
0
votes
0answers
42 views

Research in Non Commutative geometry [on hold]

I am currently doing my Masters' in Mathematics and I wish to pursue Ph.D. . I have taken courses in Differential Geometry of Manifolds and C* Algebras, and some introduction to Riemannian geometry. I ...
6
votes
1answer
140 views

Map between $SO(n)$ is homotopic to the identity?

I'm given an exercise, in a differential geometry class, where I need to detemine wether or not the smooth map between manifolds: \begin{align} f \colon\ &SO(n) \rightarrow SO(n)\\ & A \mapsto ...
2
votes
1answer
18 views

Choosing a vector normal to a jordan curve that points “inside”

Let $\gamma=\partial K_1(0,0)$ be the circle with radius $r=1$ and origin $(0,0)$ in $\mathbb R^2$. Then for any $t_0$ we have $\gamma'(t_0)\neq \begin{pmatrix} 0 \\ 0\end{pmatrix}$. Let ...
0
votes
1answer
29 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
3
votes
1answer
26 views

Condition equivalent to a moduli space being a manifold

Let $M$ be an $n$-manifold, and assume that it is foliated by a regular $p$-foliation. I know the following implication to be true: If for every point $m\in M$ there exists a submanifold $m\in ...
0
votes
3answers
41 views

The Euclidean Metric on $\mathbf R^3$ Induces an Index-Lowering Isomorphism $b:\mathfrak X(\mathbf R^3)\to \Omega^1(\mathbf R^3)$.

In Lee's Introduction to Smooth Manifolds, Second Edition, the line just before Equation 14.25 reads The Euclidean metric on $\mathbf R^3$ induces an index-lowering isomorphism $b:\mathfrak ...
2
votes
0answers
22 views

Describe the local flux of this vector field $X$ on $S^2$ given by $X(v)=w_0- \langle v, w_0 \rangle v.$

Let be $w_0 \in \mathbb{R}^3$ and $X: S^2 \rightarrow TS^2$ the vector field on $S^2$defined by: $$X(v)=w_0- \langle v, w_0 \rangle v.$$ ($\langle.,. \rangle$ is the standard dot product) How can I ...
0
votes
1answer
25 views

Four Vertex Theorem

In Do Carmo's "Differential Geometry of Curves and Surfaces" he has a proof of the four vertex theorem that I am having trouble getting my head around. In it, he starts by assuming a closed, simple, ...
5
votes
1answer
52 views

Lie bracket and flows on manifold

Suppose that $X$ and $Y$ are smooth vector fields with flows $\phi^X$ and $\phi^Y$ starting at some $p \in M$ ($M$ is a smooth manifold). Suppose we flow with $X$ for some time $\sqrt{t}$ and then ...
1
vote
2answers
73 views

Wrong pushforward of vector field definition on wikipedia

Wikipedia claims that for $$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,$$ we have for $X \in T_pM$ and $f \in C^{\infty}(N,\mathbb{R})$ $$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi)$$ whereas I ...
0
votes
1answer
21 views

Diffeomorphism between covering spaces

Let $\pi_1: M \rightarrow M_1$ and $\pi_2: N \rightarrow M_2$ be two smooth covering maps. Now $\phi: M \rightarrow N$ is a smooth diffeomorphism. Does this induce a smooth diffeomorphism $f: M_1 ...
1
vote
0answers
32 views

Is there a relation between vectors on these two spaces?

I've been reading lately one paper on Physics, which basically presents one gauge theory approach to the problem of swimming at low Reynolds number. I've been trying lately to rewrite some of the ...
1
vote
1answer
34 views

Calculating euler characteristic and geodesic curvature

We have the usual formula for the euler characteristic in differential geometry $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma g^{1/2}R + \frac{1}{2\pi}\int_{\partial M}ds k$$ where we define the ...
1
vote
0answers
38 views

Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
4
votes
0answers
55 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
0
votes
1answer
37 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
0
votes
1answer
38 views

$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...
2
votes
1answer
16 views

Is a principal bundle automorphism locally given by a left action?

Let $G\hookrightarrow P \xrightarrow{\pi} M$ be a principal bundle, denote by $\cdot$ the right action of $G$ on $P$. Let $f:P\rightarrow P$ a bundle automorphism (i.e. $f$ is a diffeo, $f(p \cdot g) ...
1
vote
0answers
25 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
4
votes
0answers
36 views

For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic

PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1 The following is my idea: First, choosing an arbitrary open coordinate ...
0
votes
1answer
15 views

Is my example of non equivalent maps correct?

We define two smooth maps $f: (\mathbb R, 0) \to (\mathbb R^2, 0)$ and $g: (\mathbb R, 0) \to (\mathbb R^2, 0)$ to be equivalent if there exist diffeomorphisms $\tau : \mathbb R \to \mathbb R$ and ...
5
votes
1answer
63 views

Is this operation meaningful or it is a mistake in the book?

I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is ...
1
vote
1answer
53 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
4
votes
1answer
48 views

Can a volume form on a submanifold be extended to a parallel form in a neighbourhood?

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n \hookrightarrow M$ be a smooth, closed, embedded submanifold. Let $\Omega$ be the volume form of $\Sigma$. It is well-known that a volume ...
0
votes
1answer
35 views

Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
-1
votes
0answers
24 views

A manifold is a covering space over its quotient by a group action

Let $M \times G\to M$ be a properly discontinuous, free action of group $G$ on a manifold $M$. The quotient topology of the orbit space is Hausdorff. Suppose $p\in M$. How can we choose an open ...
5
votes
2answers
46 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
2
votes
1answer
48 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
0
votes
0answers
13 views

Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...
0
votes
2answers
32 views

On singular points of parallels

Say $\gamma$ is a unit speed curve and its parallel is given by $$ p (t) = \gamma (t) + d n(t)$$ where $n$ is the unit normal vector and $d$ is some scalar. I read that The parallels of a ...
2
votes
2answers
31 views

On reparametrisation of curves (sorry for trivial question but I'm confused)

I'm confused about speed and reparametrisations of curves. To illustrate my confusion please let me elaborate using the simplest example I could think of: Let $\gamma : [0, 2 \pi ) \to \mathbb R^2$ ...
0
votes
1answer
14 views

Parallels of a parameterised curve if not unit speed

I just read that if $\gamma$ is a curve given in unit speed parametrisation then the parametrisation of a parallel curve is given by $$ p(s) = \gamma (s) + d n(s)$$ where $n$ is the unit normal to ...
4
votes
0answers
48 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
1
vote
0answers
44 views

What's book that I should read? [on hold]

When I read the chapter 4 of "Three Manifolds with Positive Ricci Curvature," I got stuck. I don't know what Fourier transform variable is, what derivative of second order nonlinear partial ...
4
votes
0answers
52 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
0
votes
0answers
29 views

Derivative group action [duplicate]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
2
votes
2answers
96 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
2
votes
1answer
58 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...