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 ...
3
votes
1answer
119 views
Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?
Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
3
votes
1answer
260 views
How should I deal with this two-dimensional $\frac{0}{0}$ limit?
Here is my question:
Does the following limit exist?
$$
\lim_{y\to\xi}\frac{(\xi_i-y_i)(\xi_j-y_j)({{\xi}-y})\cdot n(y)}{|\xi-y|^5},\quad 1\leq i,j\leq 3,\tag{*}
$$
where $S\subset{\mathbb ...
3
votes
2answers
250 views
What is the intuitive meaning of the scalar curvature R?
Background:
Let $M$ be a smooth, Riemannian manifold with metric $g$ and dimension $n$. Let $R^a_{bcd}$ be the Riemann tensor with respect to the Levi-Civita connection for $g$.
Question:
Is there ...
3
votes
2answers
215 views
Some references for potential theory and complex differential geometry
I am looking for references on two distinct (though related) topics.
Potential theory :
I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive ...
2
votes
0answers
36 views
Orientation-preserving diffeomorphism [duplicate]
Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
2
votes
1answer
63 views
Unique Perpendicular Geodesic
Let $p < q < r < s$ be real numbers. Let $l$ be the geodesic with endpoints at $p$ and $q$ and let $m$ be the geodesic with endpoints at $r$ and $s$.
(a) Prove that there is a unique geodesic ...
2
votes
1answer
45 views
Surfaces without conjugate points
I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
2
votes
0answers
47 views
Why is that quantity a constant?
Help needed! What have I done wrong here?
Given the metric $$ds^2 = dr^2+r^2d\theta^2$$
And $$R_{ij}=\nabla_i \nabla_j f(r)$$ where $\nabla_i$ is a covariant derivative, and $f=f(r)$ is a scalar ...
2
votes
1answer
130 views
$(n - 1)$-dimensional submanifold of the manifold $\mathbb R^n$
Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$.
Let $0 \neq b \in \mathbb{R}$.
Show that the surface $M = \{x\in \mathbb{R}^n \mid x^T A x = b\}$ is an $(n - 1)$-dimensional ...
2
votes
1answer
101 views
pullback of 1 form.
Let $i:S^2 \to \mathbb{R}^3$ be inclusion. Let $\omega=dz$ be a 1-form on $\mathbb{R}^3$. I want to compute $i^* \omega$ and it vainishes exactly two points.
Is it right that $i^* ...
2
votes
2answers
91 views
Equivalence of two definitions of differentiablitity on Regular Surfaces
When dealing with differentiable surfaces one defines a function $f:S\rightarrow \mathbb{R}$ as being differentiable if its expression in local coordinates is differentiable. But one could also define ...
2
votes
1answer
92 views
Showing: point of polytope which maximizes the minimum distance to a vertex is a barycentre?
Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, ...
2
votes
0answers
91 views
A series of Lemmas about $C^{\infty}$ vector fields
Warner page-37:
Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve ...
2
votes
1answer
180 views
are non-degenerate critical points always isolated?
I have a question regarding the isolation of critical points of a function:
Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is a $C^\infty$ function such that $f$ has a non - degenerate critical point at ...
2
votes
1answer
64 views
normal bundle of a boundary
let $X$ and $Y$ be compact, oriented manifolds and assume that $\partial X=Y$. Is it true that the normal bundle of $Y$ in $X$ is trivial? if it is the case, is there a simply explaination?
Thanks
2
votes
0answers
157 views
How to construct a vector field without zero on an open manifold?
a friend asked me to pose the following problem:
It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler ...
2
votes
2answers
152 views
Skew line dense?
Please let me refer you to:
Example 4.18. The skew line $f: \mathbb R \to S^1 \times S^1$
$$
f(t) = (e^{it}, e^{i\alpha t}).
$$
If $\alpha$ is irrational then the image of $f$ is dense in ...
2
votes
0answers
201 views
Differential forms and a chain rule
Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$.
Let $Q\in U$ ...
1
vote
2answers
225 views
Determine the result of parallel translating the vector $(0, 0, 1)$ once around the circle $x^2+ y^2=a^2, z=0$
can any one solve this problem
Q)
Determine the result of parallel translating the vector $(0, 0, 1)$ once around the circle $x^2+ y^2=a^2, z=0$ .On the right circular cylinder $x^2 + y^2 ...
1
vote
0answers
38 views
Smoothing corners of a handle attachment
Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
1
vote
1answer
117 views
Diffeomorphism on the torus
Let $S : \mathbb{R}^n → \mathbb{R}^n$ be linear invertible map, then $S$ projects to $\mathbb{T}^n$ diffeomorphism
if and only if $S ∈ GL_n(\mathbb{Z})$.
I can't prove the right to left implication.
1
vote
1answer
169 views
Covariant derivative and surface gradient
The surface gradient of a function defined on a surface $\Gamma \subset \mathbb{R}^n$ is defined
$$\nabla_{\Gamma} f = \nabla f - (\nabla f \cdot N)N$$
where $N$ is the unit normal on $\Gamma.$
How ...
1
vote
1answer
60 views
Notation and naming for two operations with $p$-form valued $n$-forms
While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much.
I'm ...
1
vote
1answer
124 views
Continuity equation on manifolds
Mass conservation is usually written as
$$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho \boldsymbol v) = 0$$
$\rho$ is the density and $\boldsymbol v$ is the fluid velocity. My attempt ...
1
vote
1answer
200 views
A difficult question about diffeomorphism about submanifold
Let $M$ and $N$ be two smooth manifolds, and $f: M \to N$ be a submersion , ${{f}^{-1}}(y)$ is compact for all $y$ in $N$. Then prove for any $x$ in $N$ there is an open neighborhood $U$ of $x$ such ...
1
vote
2answers
161 views
Differential form is closed if the integral over a curve is rational number.
The following problem comes from do Carmo's book Differential Forms and Applications, Chapter 2, Exercise 4:
Let $\omega$ be a differentiable 1-from defined on an open subset $U\subset \mathbb{R}^n$. ...
1
vote
1answer
240 views
The Dimension of the Symmetric $k$-tensors
I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the ...
1
vote
1answer
97 views
Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve
Largely I want to know as to how does one say anything about the hyperellipticity or the genus of the Riemann surface by looking at the algebraic curve and its singularities.
To give a specific ...
1
vote
1answer
767 views
Maximum sum of angles in triangle in sphere
Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is ...
1
vote
1answer
163 views
The chain rule for a function to $\mathbf{C}$
Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc.
For $a$ in $\mathbf{C}$, ...
1
vote
1answer
197 views
The Dual Pairing
My understanding from the reading the Wikipedia article on Dual Pairs is that a dual pair is comprised of two vector spaces $X$ and $Y$ over a field $\mathbb{K}$ together with a nondegenerate ...
1
vote
0answers
117 views
A particular pulling back and lifting of metric
Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a ...
0
votes
1answer
67 views
Orientation preserving diffeomorphism.
I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
0
votes
1answer
76 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 ...
0
votes
0answers
58 views
Poisson bracket identities/properties [duplicate]
Possible Duplicate:
Invertible antisymmetric matrix and identities
How does $${\partial\over \partial \xi_i}M_{jk}+{\partial\over \partial \xi_j}M_{ki}+{\partial\over \partial ...
0
votes
1answer
84 views
The shrinkage of open covering
If $M$ is a $n$-dimensional manifold which is Hausdorff and satisfies second countable axiom, for each $p \in M$, there is an open neighborhood ${U_{r(p)}}(p)$ which, through a differential mapping ...
0
votes
2answers
2k views
Multivariate Taylor Series Derivation (2D)
I am trying to understand the derivation here
kiwi.atmos.colostate.edu/group/dave/pdf/TaylorSeries.pdf
I understand how first, second total differentials are derived. I do not understand how they ...
0
votes
1answer
170 views
Canonical Isomorphism Between $\Omega^2(\mathbb{R^3})$ and $\mathbb{R^3}$?
Let $\Omega^2(\mathbb{R}^3)$ represent the collection of differential 2-forms on $\mathbb{R}^3$. For this space we take as an (ordered) basis $\{dx \wedge dy, dx \wedge dz, dy \wedge dz\}$.
First ...
0
votes
3answers
258 views
points toward the center of the osculating circle (second derivate in a arc length parameter curve)
Is it true that if I have a planar curve parametrized by its arc length, then the second derivative points towards to the center of the osculating circle?
I can´t see it, but the book says that it´s ...
0
votes
1answer
207 views
If $f: M \to N$ is a smooth map between compact connected manifolds and $\operatorname{rank}{df} = \dim{N}$ then all pre-images are diffeomorphic
Let $M,N$ be compact connected manifolds, $f:M \to N$ a smooth map with $\operatorname{rank}{(df)}=\dim{N}$. Then for all points $p,q \in N$ ; $f^{-1}p$ is diffeomorphic to $f^{-1}q$.
Please help ...
0
votes
2answers
140 views
Equivalent definitions of $C^r(\Omega)$?
The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's Lectures on Differential Geometry:
Suppose $f$ is a real-valued function defined on an open set ...
-1
votes
1answer
91 views
Length of a curve on $S^2$
$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$?
$2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates?
...
