Tagged Questions
1
vote
1answer
39 views
geometrically finite hyperbolic surface of infinite volume
I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a
"geometrically finite hyperbolic surface of infinite volume"
is mentioned frequently and I am ...
1
vote
1answer
36 views
Elementary definition: what's a parallel volume-form?
This is a very elementary question,
What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric?
To find out more about the concept, what kind of topic do I need ...
1
vote
1answer
55 views
Differentiable manifold in dimension 1 and its critical point
Please, I want to know how to define a differentiable manifold in dimension 1,
and if the circle is a differentiable manifold in dimension $1$, and what is its critical point.
Thank you.
2
votes
1answer
83 views
What is a tangential gradient?
If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write:
\begin{equation}
...
2
votes
1answer
42 views
Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$
In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n}$ $X^{i}$ $\frac{\partial}{\partial x^{i}}$, $X^{i}$ $\in$$C^{\infty}(p)$.
Then
1.The differentiation of ...
8
votes
1answer
101 views
Is there a way to relate convexity to Gaussian curvature?
This is a vague question because I'm not sure what I want to ask.
An ellipsoid has positive curvature everywhere, and bounds a convex subset of $\mathbb R^3$. What I want to say now is "It seems as ...
5
votes
1answer
85 views
For a differentiable map $\Phi$ between manifolds $M$ and $W$, what is $d\Phi?$ (Aubin)
I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this:
$(\Phi_*)_P$ is nothing ...
4
votes
3answers
125 views
What is a tangent bundle? (Aubin)
Here's what I read in A Course in Differential Geometry by Thierry Aubin.
2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$
And then
2.6. Definition. Let $\Phi$ be a ...
3
votes
3answers
52 views
Definition of a tensor field
Could anybody explain to me the following:
If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$
where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative,
then $T_{ij}$ is a tensor field.
...
2
votes
4answers
339 views
Definition for Covariant Derivative
What is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus?
definition of the derivative of a function in calculus is:
$$\frac ...
5
votes
2answers
199 views
Why does the condition of a function being differentiable always require an open domain?
Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
1
vote
1answer
57 views
Question about definition of pullback as a smooth bundle map.
In Lee, there is an exercise involving the pullback that I can't understand. If $M,N$ are smooth manifolds and $F:M\to N$ is smooth, I am asked to show that the pullback $F^*$ is a smooth bundle map ...
3
votes
2answers
229 views
What exactly is a manifold?
Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth.
Does this mean that in mathematics a manifold is essentially a representation of something that ...
0
votes
1answer
125 views
a question about definition of regular surface
While I am reading Do Carmo's differential geometry,I have several questions about the definition of regular surface.
From condition 2,the author said :
"...... $x^{-1}:V \cap S \rightarrow U$ ...
6
votes
1answer
161 views
Are these definitions of submanifold equivalent?
Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold.
Here's an "obviously" correct definition of (embedded) submanifold:
Definition A. A subspace $S$ of $M$ is a ...
0
votes
1answer
124 views
Locally Euclidean
A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a ...
3
votes
0answers
68 views
Closed / embedded surface
Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
2
votes
2answers
578 views
A simple explanation of differential calculus and its link to geometry?
The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
1
vote
2answers
138 views
On the definition of jets
I have some problems with the definition of jets and it would be
great if someone could help me here:
In many books it is written, that the $r-th$ order jet $j^r_xf$ of
a smooth function $f:M ...
1
vote
0answers
152 views
Is the extra condition in this definition superfluous?
I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
1
vote
1answer
172 views
Partial derivative notation: is that a projection function?
Consider the following definition:
Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...
11
votes
4answers
744 views
Which is the “proper” definition of a geodesic curve?
I'm taking a course on differential geometry, and up until now I'd always thought that the definition of a geodesic is (loosely speaking) a curve on a surface with the minimal length between its ...
