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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How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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36 views

Flows on a compact smooth manifold.

Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values ...
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18 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
2
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1answer
38 views

Expression for codifferential in terms of interior product

Let $(M^n,g)$ be a Riemannian manifold with local orthonormal frame $\{e_1,\ldots,e_n\}$ with dual basis $\{e^1,\ldots,e^n\}$ and with Levi-Civita connection $\nabla$. It can be checked on basis that ...
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1answer
39 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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1answer
49 views

Connectedness of level sets

I have a $C^{1}$ real valued function $f$ defined on a connected manifold $M$, it doesn't have critical points, lets assume that $f^{-1}(0)$ is a (compact) connected submanifold of $M$, does that ...
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2answers
71 views

Derivative of determinant at some point

Let $c:\mathbb{R} \rightarrow \mathbb{M}_n(\mathbb{R})$ defined by $$c(t)=A e^{tB}$$ where $A\in GL(n,\mathbb{R})$ and $B \in \mathbb{M}_n(\mathbb{R})$. The question ask me to find $c'(0)$ and ...
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85 views

Topological invariants by integrals

Some topological invariants that can be found e.g. in knot theory can be represented as integrals (Example: Integral for computing the Gauss linking number). Another example is the complex plane with ...
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1answer
46 views

Computing the differental of an orthogonal projection

I am having trouble computing the differental of a map. This is the context: Let $S\subset \mathbb R^3$ be a regular surface, and fix a point $p\in S$. Let $\pi:\mathbb R^3\to T_pS$ be the ...
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1answer
34 views

Maurer-Cartan Form on orthogonal group

I'm having trouble understanding an assignment regarding the Maurer-Cartan form on orthogonal matrices: Let $\text{O}(n)\subset\text{GL}(n,\mathbb{R})$ be the matrix Lie group of orthogonal ...
5
votes
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49 views

characterization of non compact surfaces in $\mathbb{R^3}$

Is there a way to characterize non compact surfaces with constant mean and gaussian curvature. I know that if $K=0=H$ then the surface is a plane. How can I know about the others? Just to add, for ...
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2answers
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Ambiguity in definition of $C^r$ maps between manifolds

Let $M$ and $N$ be smooth manifolds with corresponding maximal atlases $A_M$ and $A_N$. We say that a map $f : M \to N$ is of class $C^r$ (or $r$-times continuously differentiable) at $p \in M$ ...
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3answers
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Definition of a Manifold from Guillemin Pollack

I have been studying differential topology from Guillemin and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
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0answers
42 views

a question about differential geometry

Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, ...
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a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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0answers
18 views

Extending a vertical vector to a vertical vector field

Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, ...
2
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2answers
37 views

Surface with $\nabla F=(0,0,0)$

In my geometry book there is a Proposition that says that: Lets M={ (x,y,z) | F(x,y,z)=0} be a $R^3$ set, and $p=(x_0, y_0, x_0)\in M$. If $\nabla F(p) \neq(0,0,0)\; \forall p \in M$ then M is a ...
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31 views

Sets that are convex in two different metrics

Let $(M,g)$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining ...
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1answer
25 views

CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?

In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to ...
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1answer
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Coefficients of first and second fundamental form if gaussian and mean curvatures are constant.

I was solving a problem when at some point, I had this question. If $K$ and $H$ are constant for a surface. Can I say something about the coefficients of first and second fundamental form? I know that ...
7
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1answer
544 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
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1answer
27 views

What is mean by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
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51 views

Extension of Sections of Restricted Vector Bundles

Edit: Changing Question: There are two questions related questions: extending a smooth vector field extending a vector field defined on a closed submanifold I'm trying to answer a question which is ...
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1answer
39 views

Left invariant vector field

Let $G$ be a Lie Group with $e$ as the neutral element. Taken $X_e\in T_e G$, define $$X(a)=(dL_a)_e X_e$$ Why this vector field is left invariant? I get confused with the notation. Thanks!
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Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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29 views

Is the analytic version of the Whitney Approximation Theorem true?

The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous map between them ...
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1answer
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on a particular expression of riemann curvature tensor

In studying Ricci flow I found that the Riemann Curvature tensor in a orthonormal frame can be written as $$Rm(\phi,\theta)=R_{abcd}\phi_{ab}\theta_{cd} , for \phi,\theta \in \Lambda^2(V).$$How this ...
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1answer
46 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
2
votes
1answer
28 views

Dilating a curved ball

Let $B$ be a ball sitting inside a manifold $(M^n, g)$. Now, let us dilate the metric $g$ to $\lambda g$, $\lambda$ being a positive number going to $\infty$. It seems intuitively true that the ...
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2answers
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Compute the Jacobi bracket

$R(x,y,z)=(x,y,z)$ and $\Theta(x,y,z)=(xz,yz,-(x^2+y^2))$ Show $[R,\Theta]=\Theta$ $[R,\Theta]=(R\cdot\nabla)\Theta-(\Theta\cdot\nabla)R$ ...
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19 views

All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
6
votes
1answer
69 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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2answers
28 views

Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
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0answers
44 views

Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
3
votes
1answer
42 views

integrating by parts on a manifold

Suppose $M$ is compact. Let $\phi$ be some smooth function, and $\beta$ an $n-1$-form. Then does integration by parts say that $$\int_M\phi d\beta=\int_Md\phi\wedge\beta?$$ If not, how does ...
6
votes
3answers
1k views

No diffeomorphism that takes unit circle to unit square

My first time posting in this forum. This is not a homework problem. I am trying to learn my own from John M. Lee Introdcution to Smooth Manifolds. In Chapter 3, there is the problem 3-4 Let $C ...
2
votes
1answer
60 views

Differential-Geometry question- Curve Theory

Let $r(s)$ be a curve parametrized by the natural parameter $s$ and for its curvature $k$ and torsion $t$ the following condition applies: $$k(s),t(s)\neq 0 $$ for every $s$. Prove that the curve ...
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If $\Gamma\subseteq Diff(M)$ is finite dimensional, when is the evaluation $\Gamma\rightarrow M$ a submersion?

Let $M$ be a smooth manifold. Say it is compact and connected. Suppose that there exists a finite dimensional submanifold $\Gamma\subseteq\mbox{Diff}(M)$ such that the evaluation map ...
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75 views

Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
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2answers
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What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
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1answer
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Another Differential Geometry-Curve Theory

This is another problem that keeps arising year after year that none is able to solve. Any help is very appreciated. Let $r(s)$ be a regular closed curve which lays in sphere $S^2$. Prove that: ...
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1answer
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differentiable structure on mobius strip

Define $M= \mathbb{R}^2/\sim$ where $(x,y)\sim(x',y')$ if $x-x'=2n$ for some integer $n$ and $y = (-1)^n y'$. Then how can I give a differentiable sturucture on $M$? Is there a general technique for ...
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Extending a function on a submanifold to the ambient manifold & proof of a property of a vector field.

$\newcommand{\wt}[1]{\widetilde{#1}}$ Hello, I just tried my hand at two exercises from John M Lee's book Riemannian Geometry and I would like to know whether my reasoning is sound or if I did ...
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1answer
137 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
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1answer
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Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
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2answers
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Geometric characterization of critical points of the Gauss map.

Let $\Sigma \subset \mathbb{R}^3$ an oriented surface by Gauss map $N: \Sigma \rightarrow S^2$. How can I find a geometric characterization of critical points of $N$?
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Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
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1answer
40 views

a tangent vector which belongs to intersection of a manifold and a subspace is tangent to their intersection?

I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace ...
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Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...