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)

6
votes
5answers
1k views

Concrete Example Illustrating the Interior Product

Let $V$ be a finite-dimensional vector space, let $v \in V$ and let $\omega$ be an alternating $k$-tensor on $V$, i.e., $\omega \in \Lambda^{k}(V)$. Then, the interior product of $v$ with $w$, denoted ...
1
vote
1answer
322 views

Fourier transform on a simple smooth 1-manifold

Assume a very simple smooth 1-manifold, with a single chart covering, What I'd like to know is, can we use and Fourier transform for functions on this manifold just as we did for the case of ...
2
votes
0answers
161 views

What kind of manifold is one with a trivial tangent bundle? [duplicate]

Possible Duplicate: Which manifolds are parallelizable? This question is all about vector bundles isomorphic to trivial ones (for the sake of completeness I've collected definitions I refer ...
10
votes
1answer
461 views

Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential form on $M$ is a smooth map $\omega : TM \to V$ such that $\omega$ restricted to any tangent space $T_p M$ is an element of the $V$-valued exterior algebra $\Lambda^n (T_p M, ...
6
votes
1answer
198 views

What is the precise relationship between connections in differential geometry and Kähler differentials?

I've been reading some differential geometry at my leisure, and I couldn't help but getting a very familiar feeling when I've read the definition of a connection: A derivation of $M$ (or in some ...
3
votes
1answer
217 views

Why does my tangent vector not lie in the tangent space?

Me again, still learning my lesson of "don't drink and derive": I have got two parametrizations of the surface $H :=\{ (x,y,z) \in \mathbb{R}^3 \, | \, z^2 = 1+x^2+y^2, \, z > 0\}$, ...
2
votes
2answers
93 views

a neighborhood of an intersection point

if a point $x$ is in the intersection of two spaces $X$ and $Y$ suppose we know explicitly a neighborhood of $x$ in $X$, can we take the same neighborhood of $x$ in $Y$. More specifically, if the ...
0
votes
2answers
138 views

Conditions for $df(X)$ to be a smooth field

Given a vector field $X$ over a smooth manifold, under what conditions over $f$ is $df(X)$ a smooth field?
3
votes
2answers
265 views

Curve on a sphere

I'd like some hint for this problem: Show that every unit-speed curve $ \alpha : I \subset \mathbb{R} \longrightarrow \mathbb{R}^3$, whose image is in the sphere of radius $R$ , has curvature ...
3
votes
3answers
874 views

Can you extend vector fields on a manifold?

I know that not necessarily you can extend a smooth vector field defined over a subset of a manifold to ALL of the maniffold, but, can you extend it at least to an open set? (Of course I'm talking ...
5
votes
2answers
1k views

Jacobian matrix rank and dimension of the image

I am having problems with the following question: what is the relation between the rank of the Jacobian matrix of f and the dimension of the image of f? (f being continuosly differentiable),is there a ...
17
votes
2answers
698 views

Relationship between the zeros of a vector field and the fixed points of its flow

I'm having a little trouble here and would appreciate some hints. Let $M$ be a compact manifold without boundary and let $X$ be a smooth vector field on $M$ with only isolated zeros. Let $\theta_t$ ...
8
votes
2answers
545 views

What is the relation between connections on principal bundles and connections on vector bundles?

I'm reading Kobayashi / Nomizu 's vol. I. I am reading about connections in principal G-bundles. After that chapter there's a chapter on (linear) connections on Vector bundles. Since we can associate ...
7
votes
0answers
290 views

Curvature and connections in principal G-bundles

Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$. We know ...
3
votes
1answer
126 views

Computing Curvatures

What are some manifolds other than products of space forms for which the various curvature quantities can be computed easily? I'm interested in odd (real) dimensions just as much even, so I'd like to ...
1
vote
1answer
75 views

convergence of a sequence of points on a manifold

Let $M$ be a manifold with an atlas $\mathbb{a}$. A sequence of points $\{x_i \in M\}$ converges to $x\in M$ if there exists a chart $(U_i,\phi_i) \in \mathbb{a}$ with an integer $N_i$ such that ...
0
votes
1answer
257 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 ...
2
votes
1answer
201 views

scalar-flat metrics

Does anyone know if there exists a scalar-flat metric on the $n$-sphere, $n>4$, such that it is not Ricci flat. This should be easy, because it seems doubtful that spheres can carry a Ricci flat ...
3
votes
0answers
83 views

Hyperbolic manifolds

Does there exist a compact hyperbolic manifold (i.e.,all sectional curvatures are -1) with the same volume as the round-sphere of the same dimension? Does such a manifold exist for any dimension ...
4
votes
1answer
143 views

Conformally immersed Riemann surfaces and foliations

I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 ...
13
votes
2answers
1k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
3
votes
1answer
809 views

Invariant proof of the Contracted Bianchi Identity

In "Riemannian Manifolds: An Introduction to Curvature," John Lee states the following lemma: Lemma 7.7 (Contracted Bianchi Identity): The covariant derivatives of the Ricci and scalar curvatures ...
5
votes
1answer
212 views

Nonintegrable almost complex structures

The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.
4
votes
2answers
150 views

Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
1
vote
1answer
138 views

Finding a metric on a tubular neighborhood of an embedded surface

The setup for my question is an embedded surface $\Sigma\to M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_\Sigma$ on $\Sigma$ , I would like to know if there is a ...
2
votes
1answer
300 views

Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature. For instance, What is the spectrum of the ...
17
votes
9answers
3k views

Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of. I was wondering if ...
27
votes
2answers
2k views

Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
12
votes
3answers
841 views

concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
0
votes
2answers
564 views

Draw a 3D parametric curve

I have many exercise about 3d parametric curve of class $C^{\infty}(\mathbb{R})$ of the form $$ \gamma(t) = \bigl( \gamma_x(t), \gamma_y(t), \gamma_z(t) \bigr)$$ (Example of curve: $\gamma(t) = ...
13
votes
2answers
483 views

Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
4
votes
1answer
296 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of the PDE textbook(e.g. Folland's Introduction to Partial Differential ...
1
vote
1answer
119 views

Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?

In Folland's Introduction to Partial Differential Equations: A subset $S$ of ${\mathbb R}^n$ is called a hypersurface of class $C^k$($1\leq k\leq\infty$) if for every $x_0\in S$ there is an open ...
1
vote
2answers
177 views

How to understand “maximal” in the definition of differentiable structure

Consider the definition of differentiable structure (Lectures on Differential Geometry, S.S. Chern): Suppose $M$ is an m-dimensional manifold. If a given set of coordinate charts ${\mathcal A} = ...
0
votes
2answers
172 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 ...
8
votes
2answers
859 views

Can “being differentiable” imply “having continuous partial derivatives”?

Consider the following theorem: Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives ...
2
votes
0answers
232 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
1
vote
1answer
74 views

Finding a path from one boundary component to the other

Let $X$ be a compact manifold with boundary $\partial X = X_0 \cup X_1$, and let $\omega$ be a volume form on $X$. Suppose $f:X \rightarrow [0,\infty)$ is a smooth non-negative function. Is it always ...
4
votes
1answer
468 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
1
vote
0answers
284 views

Question on Stokes' Theorem

Suppose $M$ is a smooth manifold and $f$ is a real valued smooth function on $M$. Set $N:=f^{-1}([0,1])$ and suppose $N$ is a compact submanifold of $M$. Let $\mu$ be a volume form on $M$ and $v$ a ...
8
votes
1answer
664 views

Piecewise smooth, non-orientable, closed-surface: a contradiction in terms, or am I going mad?

We had a lecture a few weeks back, looking at Gauss' divergence theorem, and in the definition of the theorem, it specified that the boundary of the volume under consideration, S, had to be a ...
3
votes
1answer
559 views

Smooth boundary condition implies exterior sphere condition

If $\Omega$ is a bounded domain in $\mathbf{R}^n$ with $C^2$ boundary, show that $\Omega$ satisfies "exterior sphere condition". Exterior sphere condition means that for each $z\in\partial\Omega$, ...
2
votes
1answer
334 views

exact differential n-forms

We know that a 1-form $\omega$ on a manifold $M$ is exact if and only if $\int_{\gamma}\omega=0$ for any closed loop $\gamma$. How can I prove the following generalization: $\omega$ is an exact n-form ...
5
votes
2answers
372 views

Gauss-Bonnet-Chern theorem

Good morning/day/evening/night, I was presented to the generalized Gauss-Bonnet-Chern theorem for hypersurfaces in Euclidean space; For a closed, even dimensional manifold $M$ with dimension $n$ ...
14
votes
1answer
441 views

Coordinate-free techniques in Lagrangian mechanics

Consider the following Lagrangian (Exercise 3.6B from Abraham and Marsden's Foundations of Mechanics): $$ L(\upsilon)=\frac12g(\upsilon,\upsilon)+V(\tau_Q\upsilon)+g(\upsilon,Y(\tau_Q\upsilon)) $$ ...
0
votes
0answers
121 views

homology Questions

I have some questions and would be infinitely grateful to you for your answers: 1- $f^{*}$ being the dual of $f_{*}$ so the degree (between top dimensional (co)homology groups) is the same for both ...
5
votes
2answers
600 views

What is the definition of tensor contraction?

According to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V^* ...
11
votes
1answer
1k views

precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
11
votes
1answer
333 views

What is the universal property of the tangent bundle of a smooth manifold?

The process of writing my own notes on smooth manifolds have led me to wonder about this. All I've really found is the following: In addition to Madame Ehresmann's references, there is in ...
12
votes
1answer
637 views

Car movement - differential geometry interpretation

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows: Denote by $C(x,y)$ the center of the ...