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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853 views

Expression for Levi-Civita Connection

I'm having trouble with the following exercise in do Carmo's Riemannian geometry. Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be ...
0
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1answer
491 views

Question on Partitions of Unity

I was reading John Lee's Introduction to Smooth manifolds, and I came across this question: Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous ...
3
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2answers
278 views

To what extent do the stories on manifolds carry over to schemes?

This is a follow-up (refinement?) of this question. In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of ...
7
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2answers
713 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
12
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5answers
854 views

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
12
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2answers
764 views

PDEs on Manifolds

I am wondering if there is a general coordinate-independent way to define a Partial Differential Equation on a Smooth manifold. It is definitely true that in each coordinate neighborhood you could ...
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0answers
788 views

The set of symmetric matrices as a manifold

How would I start off proving that $S= $(the set of symmetric $n\times n$ matrices) is a manifold. I tried using the definition directly by saying $M_n =$ the space of all $n\times n$ matrices For ...
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2answers
173 views

Harmonic functions on $S^2$

Consider the sphere $S^2 = \lbrace (x,y,z) :\ x^2 + y^2 + z^2 = 1 \rbrace$. This is a smooth manifold in $\mathbb{R}^3$, and for a given point $s \in S^2$, one can consider its coordinate ...
3
votes
3answers
1k views

The function that draws a figure eight

I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the ...
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3answers
239 views

A question about Gauss Bonnet theorem

If $S$ is a surface which is the complement of finitely many points in a compact surface, and the metric in $S$ is complete, then is Gauss-Bonnet theorem still valid for $S$?
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0answers
136 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 ...
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votes
3answers
1k views

Conditions that torsion is zero in a space curve

What are the conditions for torsion to be zero other than having a plane curve? The only thing I can thing of is an equation that have the torsion that cancels out each other.
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3answers
644 views

Is the scalar curvature the only isometric invariant of a Riemannian 2-manifold?

Given two Riemannian Manifolds of dimension 2, and a point on each. If the scalar curvatures are isomorphic (as functions) in some neighbourhoods of these points, are then the manifolds necessarily ...
3
votes
3answers
273 views

Riemannian 2-manifolds not realized by surfaces in $\mathbb{R}^3$?

A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$ (i.e., distance measured by shortest paths on $S$) is a Riemannian 2-manifold: differentiable because ...
0
votes
1answer
96 views

An inequality about norms of vector fields on Riemannian manifolds

Let $g_{ij}$ be the components of a symmetric rank-2 positive definite tensor (metric on a Riemannian manifold). Let ${C^i}$ and ${ \beta ^i }$ be components of a vector field on it, the former of ...
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1answer
382 views

Boundary of a 2-chain

If $c$ is a singular 1-cube in $\mathbb{R}^2 \backslash \left\{ (0,0)\right\}$ with $c(0)=c(1)$, show that there is an integer $n$ such that: $c-c_{(1,n)} = \partial (c^2)$ for some $2$-chain $c^2$.
11
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2answers
683 views

Why is the manifold structure on the tangent bundle unique?

...subject to the conditions that (i) the projection be smooth and that (ii) smooth sections correspond to smooth vector fields. This homework problem is really bugging the hell out of me. Of course ...
3
votes
1answer
283 views

Is the notion of density really needed to define integration on nonorientable manifolds?

I am trying to understand, in as simple terms as possible: How to define integration for non-orientable manifolds, and why it is impossible to do so using only differential forms. In particular, ...
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3answers
1k views

How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$?

If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\backslash \left\{ (0,0)\right\}$, then how do I show/prove that $d\theta =\dfrac{x dy - y dx}{x^2 + y^2}$?
4
votes
3answers
1k views

Some questions about differential forms

If $A$ is a differential one form then $A\wedge A .. (more\text{ }than\text{ }2\text{ }times) = 0$ Then how does the $A\wedge A \wedge A$ make sense in the Chern-Simon's form, $Tr(A\wedge dA + ...
7
votes
1answer
287 views

When does the topological boundary of an embedded manifold equal its manifold boundary?

Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the ...
9
votes
2answers
221 views

Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?)

This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering ...
4
votes
1answer
499 views

What is the modulus of a tensor on a Riemannian 3-manifold?

Let $v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let ...
7
votes
1answer
309 views

In which commutative algebras does any derivation possess a flow?

Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\gamma\colon ...
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0answers
247 views

A particular method of pulling back a metric on a submanifold

Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using ...
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vote
2answers
302 views

A nice proof of the fact that the Quadratic Term of the Taylor polynomial is Curvature

For plane curves, twice the quadratic term of the Taylor expansion at a point on that curve is precisely the curvature. I know of one proof as featured in Hubbard and Hubbard, but I was wondering if ...
7
votes
1answer
412 views

A good lower bound on the maximum curvature in a loop

Suppose $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ is a $C^\infty$ curve, parameterized by arc length ($\left\|\alpha'(t)\right\| = 1$), and with $\alpha(0) = \alpha(\ell)$. Show that there exists ...
11
votes
4answers
4k views

Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
4
votes
2answers
493 views

Parallel transport of a vector along two distinct curves

Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same ...
3
votes
1answer
185 views

Terminology for point in dent in surface?

This is a simple terminology question. Let $S$ be a (let's say smooth) surface in $\mathbb{R}^3$, and $p$ a point on $S$. Suppose the principle curvatures $\kappa_1$ and $\kappa_2$ at $p$ are both ...
7
votes
1answer
579 views

Kernel of the tangent map

If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that if ...
5
votes
2answers
263 views

How are gauge transformations of a $G$-bundle related to the adelic points of $G$?

In a very interesting blog discussion at the $n$-category cafe, an anonymous poster made the following remark: "... using the dictionary between number fields and function fields, Weil suggested that ...
2
votes
1answer
242 views

Does this surface minimize maximum distance?

Suppose we have a tetrahedron defined by points (0,0,0),(1,1,0),(0,1,1),(1,0,1). Now define surface by (a,b,a + b - 2*a*b) for a,b between 0 and 1. Let E1 be the set of points inside the tetrahedron ...
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vote
1answer
315 views

Euler characteristic of a compact surface

Determine the Euler characteristic of the surface $$ M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} $$
5
votes
1answer
734 views

Converse To Quotient Manifold Theorem [Exercise in Lee Smooth Manifolds]

I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one] If a Lie group $G$ acts smoothly and freely on ...
5
votes
1answer
301 views

Principal bundles on 3-manifolds

If G is a simply connected Lie Group then why is every G-bundle over an orientable 3-manifold trivial? (Why is orientability important?)
2
votes
2answers
867 views

Taking trace of vector valued differential forms

Can anyone kindly give some reference on taking trace of vector valued differential forms? Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true? ...
4
votes
1answer
467 views

Gauge transformations in differential forms

I am aware of gauge transformations and covariant derivatives as understood in Quantum Field Theory and I am also familiar with deRham derivative for vector valued differential forms. I thinking of ...
10
votes
1answer
348 views

What is the theory of non-linear forms (as contrasted to the theory of differential forms)?

It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not ...
2
votes
1answer
111 views

Tolman-Bondi-Lemaitre space times

One can see this reference for TBL space-times. I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained. ...
3
votes
2answers
352 views

How to compute curvature tensors for general n-dimensions?

I keep coming across calculations like this, Consider a metric on an $n+2$ dimensional manifold given as, $ds^2 = 2dudr + 2L(u,r)du^2 -r^2d\Omega_n^2$ Then apparently once can write down the Ricci ...
9
votes
2answers
609 views

Are there higher-dimensional analogues of sectional curvature?

I recently learned that on Riemannian manifolds, one can define the sectional curvature (http://en.wikipedia.org/wiki/Sectional_curvature) of a (2-dimensional) plane section. I was wondering if a ...
4
votes
1answer
829 views

Explicitly proving invariance of curvatures under isometry

I would like to know how to explicitly prove that Riemann Curvature,Ricci Curvature, Sectional Curvature and Scalar Curvature are left invariant under an isometry. I can't see this explained in most ...
2
votes
1answer
207 views

Local equation for the manifold from its principal curvatures

If $k_1,k_2,...,k_n$ are principal curvatures of a hypersurface in $\mathbb{R}^{n+1}$ then one can apparently locally parametrize the manifold as $(x_1,x_2,...,x_n,y)$ such that, $y = \frac{1}{2}(k_1 ...
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votes
2answers
251 views

Books on Lie Groups via nonstandard analysis?

Are there any books or online notes that cover the basics of lie groups using nonstandard analysis? Another thing I would like is a to see these things set in category theory (along the lines of ...
5
votes
1answer
236 views

What is the form of curvature that is invariant under rotations and uniform scaling

This is a followup to this question, where I learned that curvature is invariant to rotations. I have learned of a version of curvature that is invariant under affine transformations. I am ...
5
votes
1answer
357 views

Is a curve's curvature invariant under rotation and uniform scaling?

The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?
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1answer
2k views

Intuitive explanation of covariant, contravariant and Lie derivatives

I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results. ...