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 ...

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113 views

Zero Locus of Functions is a Submanifold

Suppose $f_1,\dots, f_d$ are a set of real-valued functions on a smooth manifold $M$. Let $N$ be the zero locus of the $f_i$. Suppose the $\textrm{d}f_i$ span a subspace of the cotangent space of $M$ ...
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3answers
416 views

Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
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0answers
170 views

why we cannot integrate on a nonorientable manifold?

I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a ...
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48 views

List of minimal surfaces embedded in the 3-sphere

Is the set of possible areas of closed, embedded, minimal surfaces of the 3-sphere discrete?
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1answer
40 views

Deduce the behavior of the following $f(s)$ and $\gamma(s)$

Let $\gamma(s)$ be a smooth curve in $\mathbb{R}^3$ parametrized by arclength. Supposed that for some function $f(s)$, $\gamma''(s) = f(s)\gamma(s)$. What can you deduce about $f(s)$ and $\gamma(s)$? ...
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2answers
152 views

Deciding whether a given set is a manifold

This question should be extremely elementary but it stumped me somewhat. I know the definition of a (topological/smooth) manifold but I seem to have trouble when it comes to deciding whether a given ...
3
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1answer
68 views

a question about germs of functions

Let $M$ be a smooth ( real) manifold, if $ p\in M$ and $f\in C^{\infty}(U)$ ($U$ is an open subset of $M$), the symbol $[ f]_p$ indicates the smooth germ of $f$ at $p$ . Consider the following set ...
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1answer
363 views

isometry and exponential map

I got stuck on the following questions. Can anyone give me idea how to proceed? Suppose $M$ is a Riemannian manifold and $\phi: M \to M$ an isometry map. If $\phi(p)=p$ and $\phi(q)=q$ prove that ...
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2answers
56 views

Is the following curve $\gamma(s)$ closed?

Let $\gamma(s)$ be a curve (parametrized by arclength) whose image lies on the circular cylinder $x^2+y^2=1$ in $R^3$, given that curvature $\kappa(s)>0$ and that torsion $\tau(s)=0$ for all $s$. ...
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1answer
182 views

characterizing semi-Riemannian spaces of constant curvature

How does one characterize $n$-dimensional semi-Riemannian spaces of constant curvature? By "characterize," I mean giving both a definition and some insight into how the possibilities work out in ...
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1answer
58 views

Transversality of a mapping

The question I have is: Show that the mapping $g:R^2 \rightarrow R^3 $ given by : $y_1 = x_1 (x_1 ^2 -x_2 ^2 +1), y_2=x_2, y_3=x_1 ^2$ is transversal to all lines $y_2 = \textrm{constant}$ in the ...
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560 views

Differential Geometry on the Sphere: Integration by Parts, Volume Form,Stokes Theorem

I was trying to work out a integration by parts formula $S^2$ of the form $$\int_{S^2}f(x)\frac{\partial g}{\partial x_1}dx \tag{1}$$ where $f,g:\mathbb{R}^3\rightarrow\mathbb{R}$. Given $g$ and ...
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1answer
210 views

Is the tangent bundle the DISJOINT union of tangent spaces?

Let $M$ be a smooth manifold and consider the Lee's definition of the tangent space $T_pM$ (so $T_pM$ is the vector space of derivations at $p$). The canonical definition of tangent bundle (as set) of ...
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2answers
352 views

Extending a Set of Linearly Independent Vector Fields to a Basis

My question is this. Suppose we are given some smooth vector fields $X_1, X_2,..., X_k$ which are linearly independent at all points in a neighborhood $U$ (EDIT: diffeomorphic to a ball) of $R^n$. Do ...
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2answers
777 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
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0answers
572 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
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1answer
181 views

Invariant vector field by group action

in a solved exercise, there is a point in the solution that I can't work out. I would be grateful if somebody could give me the detailed steps. Consider the trivial principal bundle $P = M\times ...
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0answers
246 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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3answers
521 views

Normal subgroup and Lie algebra

I have an exercise of Lie group as follows: "Let $G,H$ be closed connected subgroup of $GL_n(\mathbb{R})$, and $H$ be subgoup of $G$. Suppose that $Lie(H)$ is an ideal of $Lie(G)$. Prove that $H$ is a ...
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2answers
114 views

Minimal surfaces and gaussian and normal curvaturess

If $M$ is the surface $$x(u^1,u^2) = (u^2\cos(u^1),u^2\sin(u^1), p\,u^1)$$ then I am trying to show that $M$ is minimal. $M$ is referred to as a helicoid. Also I am confused on how $p$ affects the ...
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618 views

What would be the shortest path between 2 points when there are objects obstructing the straight path?

How would an algorithm find the shortest distance between 2 points on a horizontal 2d plane , especially when a straight path is not possible? Could it be something on the lines of calculating least ...
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2answers
107 views

Is $V\otimes V\otimes V^*$ and $V\otimes V^*\otimes V$ the same?

Are $V\otimes V\otimes V^*$ and $V\otimes V^*\otimes V$ the same? I think tensor product is commutative and associative, so I think they are the same thing, then why is it necessary to use different ...
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2answers
733 views

There is no isometry between a sphere and a plane.

How can I show that there is no isometry between a sphere and a plane? Wikipedia defines an isometry as follows: Let $(M,g)$ and $(M',g')$ be two Riemannian manifolds, and let $f:M\to M'$ be a ...
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2answers
82 views

Defining an f-invariant measure

Suppose I have a compact oriented manifold $M$ with an orientation preserving self-diffeomorphism $f$. I wish to define a volume form on $M$ which is invariant under $f$. Certainly, it is necessary ...
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72 views

show that subset of complex projective space is a submanifold

Let n, m $\in \mathbb{N}$. I'm trying to show that $M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$ is a submanifold and codim(M(n,m))=2. My idea was to use ...
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0answers
53 views

Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos ...
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1answer
127 views

Does diffeology provide moduli for classical constructions?

Do classical constructions on differentiable manifolds like affine connections, Riemannian metrics, or (almost) complex structures have moduli spaces in category of diffeological spaces?
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1answer
49 views

is a non-falling rank of smooth maps an open condition?

If $f \colon M \to N$ is a smooth map of smooth manifolds, for any point $p \in M$, is there an open neighbourhood $V$ of $p$ such that $\forall q \in V \colon \mathrm{rnk}_q (f) \geq \mathrm{rnk}_p ...
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67 views

Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
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1answer
36 views

Why this map is identically $0$?

I encountered following proposition: Let $p$ be a prime, $R=\mathbb{Z}/p^2\mathbb{Z}$, $M=\mathbb{Z}/p\mathbb{Z}$ as $R$-module. Then we have the map $M^*\otimes_RM\rightarrow$Hom$_R(M,M)$ given by ...
3
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1answer
55 views

Linear map between $M^*\otimes_RN\rightarrow \text{Hom}_R(M,N)$

I'm reading a lecture note about tensors, following is a proposition: For $R$-module $M$ and $N$, there is a linear map $M^*\otimes_RN\rightarrow \text{Hom}_R(M,N)$ sending each elements ...
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1answer
518 views

Prove the regular surface with 2 geodesics from p to q, and negative curvature cannot be simply connected.

What ideas/formulas are required to solve this? Exercise: If a and b are two geodesics from point p to q, how do you prove that M is not simply connected? M is a regular surface in R3 and has negative ...
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1answer
103 views

Given a closed submanifold $Y$ of $X$ and a $C^{\infty}$ map $f$ on $Y$, can $f$ be extended to $X$?

I want to extend $f$ locally in the intersection of each coordinate patch of $X$ with $Y$, (and set it to $0$ outside of $Y$) and then use a partition of unity to get a differentiable map that agrees ...
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1answer
166 views

When does the normal vector of a Moebius-strip intersect?

In class the teacher was talking about normal vectors. $r = \langle x,y\rangle$ then the normal vector is $$ N\left(t\right) = \frac{T^{\prime}\left(t\right)}{||T^{\prime}\left(t\right)||} $$ where ...
9
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1answer
572 views

intrinsic proof that the grassmannian is a manifold

I was trying to prove that the grassmannian is a manifold without picking bases, is that possible? Here's what I've got, let's start from projective space. Take $V$ a vector space of dimension n, and ...
10
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2answers
245 views

Almost A Vector Bundle

I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing ...
12
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1answer
477 views

Differentiable manifolds as locally ringed spaces

Let $X$ be a differentiable manifold. Let $\mathcal{O}_X$ be the sheaf of $\mathcal{C}^\infty$ functions on $X$. Since every stalk of $\mathcal{O}_X$ is a local ring, $(X, \mathcal{O}_X)$ is a locally ...
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1answer
50 views

How can we find the arc length of the curve? [closed]

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$
3
votes
2answers
180 views

Explanation about frames as distinct from a co-ordinate system

I am quite confused as to what is the difference between a frame and a co-ordinate system. The wikipedia page was not very helpful for me. I would be very happy if someone could give me a non-rigorous ...
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0answers
71 views

Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
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0answers
36 views

Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?

As the title saying, Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module? Here we are considering vector bundles with finite-dimensional fibers.
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2answers
390 views

Hodge dual on orthonormal basis: two inconsistent answers

I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
2
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1answer
248 views

How show that a surface embedded is non-orientable?

Let be $M$ a compact $3$-manifold. If $\Sigma$ is a embedded surface in $M$, such that $\Sigma$ is homeomorphic to $\mathbb{RP}^2$. If $i: \pi_1(\Sigma) \longrightarrow \pi_1(M)$ is not injective, ...
0
votes
1answer
74 views

Is any continuous curve in $\mathbb{R}^n$ a 1-D manifold?

I wonder if there is any theorem stating that any continuous curve in $\mathbb{R}^n$ is a 1-D manifold. If not, can anyone provide an example? At first I thought maybe a Peano curve affords a ...
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3answers
628 views

Example of non-manifold surface.

Is there any example of a surface which is locally homeomorphic to $R^n$ but is not a manifold? (i.e. does not have an well-defiend atlas)
2
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0answers
106 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
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2answers
152 views

Identifying functions on the unit disk with functions on the upper hemisphere

I've been wondering about something, and it might be nonsense (if so I apologize!). Consider the unit disk in $\mathbb{R}^2$ and a function $f$ defined on the disk. I can compute its double integral ...
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280 views

What is the intuition behind the Lie derivative of a vector field.

We have the following two formula about the Lie derivative of a vector field: $$ \left.\frac{d}{dt}\right|_{t=0}T\varphi_{-t}\cdot Y_{\varphi_t(p)}=[X,Y]_p = (\mathcal{L}_XY)(p) $$ where ...
4
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1answer
63 views

Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature

So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve ...
3
votes
1answer
372 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...