For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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1answer
83 views

$V$ vector field, $\omega$ one-form, $V(\omega(V))$=?

(1-forms) Let $X$ be a manifold and $\omega \in \Omega^1(X)$ be a smooth 1-form, and $V, W \in V^{\infty}(X)$ smooth vector fields on $X$. Then, $\omega(V ), \omega(W ) \in C^{\infty}(X)$ are ...
4
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0answers
58 views

Lie Bracket and vector fields

Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such ...
1
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1answer
44 views

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$.

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$. My try : By definition of derivative of a function $f:\mathbb{R^n} \to \mathbb{R^m}$ , If I know ...
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1answer
24 views

Is a one-form a derivation on $C^ \infty$?

I know that a vector field is a derivation on $C^ \infty$, meaning that it is R-linear and Leibnizian. Is it the same case for one-forms?
4
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0answers
32 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
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0answers
27 views

How to find a curve inside a non-convex

I want to connect two points in a space within the space. If the space is convex, I can simple draw a line between them. But how about a non-convex space. How can I find a curve connecting these two ...
3
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1answer
59 views

Geometric interpretation of Supersymmetry

Is there a geometric interpretation of supersymmetry? I.e., if one has a manifold $\mathcal {M} $, and there are $\mathcal {N} $ SUSY generators, then is there a geometric interpretation of the SUSY ...
2
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0answers
55 views

Local dimension of graph embedding

I am trying to find a way to characterize the dimension of the smallest space into which a (neighbourhood of) a graph $\Gamma = (V, E)$ may be embedded. Although in the end my goal is to identify what ...
5
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1answer
85 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
1
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0answers
35 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
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2answers
24 views

Multilinearity of the exterior derivative of a one-form.

I wish to show that the exterior derivative $d \theta$ of a one-form $\theta$ is $\frak{F} \mathrm{(M)}$-multilinear, therefore, a tensor. Let $X, Y, V, W \in \frak{X} \mathrm{(M)}$ and $f, g, h, k ...
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1answer
126 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 ...
2
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1answer
61 views

How to decide if a given set is a manifold

Could you tell me how to decide if a certain set is a manifold? I know there already is a similar question here, but there we have fairly "visualizable" sets: a hemisphere and a square. What in ...
0
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1answer
34 views

Vanishing of a covector (1-form) and a vector field

a) A one-form $\theta$ is zero if and only if $\theta X = 0$ for all $X$ in the set $\frak{X} \mathrm{(M)}$ of all smooth vector fields on a manifold $M$. b) A vector field $X$ is zero if and only ...
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1answer
60 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
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0answers
46 views

Is a manifold over $\mathbb{R}$ normal?

We have manifold $G$ over the reals with its finite atlas ($g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}, G=\bigcup U_i$). The atlas induces a topology in the normal way ($A \subseteq G$ is open iff ...
2
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2answers
70 views

Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
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0answers
42 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!
3
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1answer
77 views

Partial derivative of a function on manifold

Bishop and Goldberg define ("Tensor analysis on manifolds") the partial derivative of a smooth function on a manifold $M$ in the following way: $\partial_i f= \frac{\partial ...
0
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0answers
51 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 ...
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1answer
105 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
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1answer
53 views

Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
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2answers
70 views

Is a closed compact 2-Manifold that is embedded in euclidean 3-space always orientable?

I am sorry if this is a trivial question but I am a little confused right now so please bear with me. Since non-orientable compact 2-manifolds without boundary cannot be embedded in three-dimensional ...
2
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2answers
66 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)?
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1answer
44 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 ...
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0answers
34 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^N$. Can every question about ...
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0answers
31 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 ...
1
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1answer
63 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!
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0answers
19 views

Tangent Space to the Set of Congruent Points

I have been solving a question for a long time and it is finally starting to give. Definitions and Progress We use $(\mathbb R^n)^m$ to denote $\overbrace{\mathbb R^n\times\cdots\times\mathbb ...
2
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1answer
48 views

Integration on compact manifold

Integration on a nice enough manifold of a function $f:M \to \mathbb{R}$ is defined $$\int f = \sum_{ i \in I} \int_{U_i}\phi_i f$$ where $\phi_i$ is a partition of unity subordinate to the open cover ...
3
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1answer
65 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
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2answers
69 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 ...
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1answer
266 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
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1answer
94 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 ...
1
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1answer
32 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
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1answer
71 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 ...
0
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1answer
49 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 ...
3
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3answers
29 views

Is a non-orientable surface a kind of manifold?

An example of a non-orientable surface is the Moebius Strip. Iam just curious whether it is indeed a manifold by definition.
0
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0answers
29 views

A smooth map $\phi:M\rightarrow N$ is smooth if and only if $g \in C^{\infty}\left(N\right)$ implies $g\circ\phi \in C^{\infty}\left(M\right)$.

This is a problem from Semi-Riemannian Geometry by O'Neill. I can easily prove that if $\phi:M\rightarrow N$ is smooth and if $g \in C^{\infty}\left(N\right)$ then $g\circ\phi\in ...
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2answers
73 views

Does a smooth mapping always have an inverse map which is also smooth?

If not, can someone provide counterexamples? Thank you
2
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2answers
84 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 - ...
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0answers
12 views

Persistence of invariant manifolds under noninvertible mappings

According to Fenichel's classic result, a normally hyperbolic invariant manifold of a diffeomorphism persists under perturbations to the diffeomorphism. Normal hyperbolicity requires contraction or ...
1
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1answer
89 views

Set of matrices differentiable manifold? [closed]

Let $X$ be a set of matrices $2\times 3$, that for all $A$ from $X$ rank $A=1$. Is $X$ a manifold? If not find a maximum subset in $X$, which is a manifold and its dimension.
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0answers
83 views

Gauss map and Gaussian curvature of the generalized cone.

Show that the image of Gauss map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.
0
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1answer
45 views

Triangle a manifold

Let $x,y,z \in \mathbb{R}^3$ and $\Delta:=\text{conv} \{x,y,z\}$ be a triangle. My question is: Is this triangle a $C^2$ submanifold in $\mathbb{R}^3$? The reason is, that I would need this fact in ...
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0answers
13 views

Looking for a basic reference on propagators (in Topology)

I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks!
2
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1answer
44 views

Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
1
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1answer
49 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
0
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1answer
63 views

Derivation for affine connection formulas on differentiable manifolds (General tensors)

Let $p\in U\subseteq M$ be a point in some neighborhood of a finite-dimensional differentiable manifold, $\{x^i\}$ a set of local coordinates with respect to $U$, and ...
0
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1answer
35 views

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature.

Show that the image of Gaussian map of a generalized cone is a curve on $S^2$ and deduce that the cone has zero Gaussian curvature. I dont have enough idea. Please explain the question clearly. ...