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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Are there closed manifolds that can be given multiple geometric structures of constant curvature

For example, is there a closed differentiable manifold that can be given both a Euclidean structure and a Hyperbolic structure? If not, is there a reasonably easy proof that no such manifold exists?
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59 views

Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
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1answer
112 views

Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
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1answer
40 views

A question on the differential of the Gauss map

Let $S$ be a orientable, regular surface, locally parametrized by $(U, F, V)$. Let $N$ be the Gauss map. The Weingarten map is defined with a point $p$ in $U$ as $W_p: T_pS \rightarrow T_pS$ , ...
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40 views

Foliation preserving diffeomorphisms for a codimension 1 foliation

I am studying reference frames on Minkowski spacetime $\mathcal{M}$, with (+,-,-,-) signature, from a differential geometric point of view, for this reason I came up with (codimension 1) foliations ...
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1answer
27 views

Why is this statement about integral curves correct?

The following definitions and example are taken from John Lee's Smooth Manifolds, 2nd edition. Given a vector field $V$ on a smooth manifold $M$, we define an integral curve of $V$ to be a ...
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1answer
33 views

Alexander duality formulation + Jordan-Brouwer separation

In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality $\tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)$? (2) Is it assumed that the ...
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1answer
63 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
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2answers
112 views

Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
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1answer
77 views

What does it mean for two hyper-surfaces to be tangent to each other?

In the book "Anathem" by Neal Stephenson, Part four begins: Six weeks after I joined the Edharian order, I became hopelessly stuck on a problem that one of Orolo's knee-huggers had set for me as a ...
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38 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold? Explanation Suppose we have a 3D smooth manifold ...
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1answer
116 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
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1answer
47 views

Find parametric equations of a curve osculating plane at a point…

Find parametric equations of a curve osculating plane at a point corresponding to the parameter value $t$ is: $$x + ty + t^2(y) = t^4 $$ I have read the textbook and I can't find any examples.
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4answers
135 views

Proving $\dfrac{dN}{ds}=-\kappa T+\tau B$

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have ...
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1answer
120 views

Totally geodesic and autoparallel

Let $M$ be a Riemannian manifold. A submanifold $N$ of $M$ is totally geodesic if every geodesic in $N$ is also a geodesic in $M$. On the other hand, $N$ is an autoparallel submanifold of $M$ if ...
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1answer
49 views

When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
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45 views

curvature$=0$ implies straight line?

The fundamental theory of differential geometry states that: If there is a given curvature $\bar{\kappa}(s)>0$ and torsion $\bar{\tau}(s)$ which both of them are differentiable and continuous in ...
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69 views

principal axis of a volume from moments of inertia

I'm trying to calculate the expression to find the principal axis of a volume via its moments. In the 2D case I can formulate the problem by expressing the moments around arbitrary axes $x' = x \cos ...
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1answer
90 views

parallel curvature imply constant Ricci and scalar curvature

$\text{Suppose we have} \nabla R = 0 $, where R represents curvature tensor, Prove that Ricci curvature and scalar curvature are constant.
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28 views

References for Conjugate Points in Differential Geometry

I will have to give some lectures about conjugate points and I need some nice references about it, can anyone recommend me some? I already know manfredo's differential geometry of curves and surfaces ...
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1answer
41 views

How to show $\displaystyle \kappa=\frac{\langle \alpha^{''}, J(\alpha^{'})\rangle}{\langle \alpha^{'}, \alpha^{'}\rangle^{3/2}}$?

Let $\alpha(t)=(x(t), y(t))$ be a regular curve (not necessarily an unit speed curve) in $\mathbb R^2$. How to show the curvature of $\alpha$ is given by $$\displaystyle \kappa=\frac{\langle ...
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2answers
95 views

Cauchy's Theorem by Differential Geometry

Is there a prove of Cauchy's theorem footing on the topology of the complex plane (homotopy, differential forms, etc.)? More specific consider a differentiable Banach space valued complex function. ...
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1answer
38 views

Central curve $\alpha^*(t)$ is uniquely determined by the following condition?

Let $\alpha:I\longrightarrow \mathbb R^2$ be a regular curve with curvature $\kappa>0$ and normal vector $N$. I need some help to show the following: Show the curve, ...
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28 views

A question about a surface

This is a question in "baby do Carmo" 2.2.14. The question says A half-line $[0, \infty)$ is perpendicular to a line $E$ and rotates about $E$ from a given initial position while its origin $0$ moves ...
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66 views

Some detail in interior product

This is a content in page 35 of foundation in differential geometry - KN For a form $r$-form $\omega $ interior product is $$ i_X\omega\doteq C(X\otimes \omega)$$ where $\{ e_i\}$ is ON, notation ...
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50 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
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24 views

$[D,D']$ where $D$ is a derivation and $D'$ is skew

This is a proposition in 33 page of Foundation in Differential Geometry - KN I need some detail. Let $D^r(M)$ be a set of $r$-form. Then derivation (resp. skew-derivation) of degree $k$ is a ...
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42 views

Explicit metric of K3 manifold

As a physicist, I am currently working through String-Theory and M-Theory by Becker, Becker and Schwarz. In the string geometry chapter, they introduce Calabi-Yau manifolds, and in particular the K3 ...
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60 views

Strictly convex boundary of Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. What does it mean to say that the boundary is convex and strictly convex? I can find definitions of ...
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17 views

spin^c structures and charged spinors

Given a spin structure and a complex line $\mathcal{L}$ we can form the tensor product of the complex spinor bundle $S$ and this line $S\otimes\mathcal{L}$. A spin^c structure attempts to construct ...
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132 views

Periodicity of Ideal Packings on Arbitrary Boundaries

This is a question that's been nagging me for some time that I find particularly interesting, but have no idea how to go about solving it. Consider some arbitrary solid $S$, such that $\partial S$ is ...
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34 views

Component of a pushover vector by one-parameter transformation

I am curious about a step on the proof that shows Lie derivative of a vector field is equivalent Lie bracket. Following comes from Nakahara. We define integral curves by vector field X and Y as ...
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1answer
46 views

Why is the set of positive definite matrices in $\mathbb R^{n\times n}$ a positive cone

The set of positive definite matrices in $\mathbb R^{n\times n}$ is geometrically a positive cone. This statement appears in almost every article on real positive definite matrices I read but without ...
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Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
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1answer
39 views

The extension of functions

Let $f$ be a smooth function defined on $[a,b]$ and $g$ a smooth function defined on $[c,d]$. If $a<b<c<d,f'(x)>0, g'(x)>0$ and $f(b)<g(c)$, then can we find a function $h: \mathbb R ...
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2answers
133 views

Computing cohomology of hypersurface

I'm taking a course on differential geometry now, and we got the following exercise from the lecturer: compute the (de Rham) cohomology groups $H_{dR}^i(M)$ of your favourite space. In all the ...
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21 views

Isotropy of transitive lie algebroid

Let $\rho:E{\longrightarrow} TM$ is a transitive lie Algebroid. I wanna show $Ker\rho$ is a vector sub-bundle of $E$ by introducing its bundle-chart. please hint me.
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178 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
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1answer
92 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
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1answer
78 views

diffeomorphism preserve a volume form

Let $\omega_1$, $\omega_2$ two volume form on a compact manifold $M$, we know that there exists a never-vanishing function $f$, s.t. $\omega_1=f\omega_2$. If $h$ is a diffeomorphism $M \to M$ ...
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32 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
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1answer
35 views

Spherical Curve Problem..

I need some help with the following problem of differential geometry: Suppose $\alpha$ is a unit speed curve with curvature $\kappa>0$ and torsion $\tau\neq 0$. $\bf (a)$ If $\alpha$ lies on a ...
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33 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
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2answers
72 views

Is a tangent to a curve in a hyperbolic plane straight?

Consider a projective plane with an absolute quadric, so that it is a hyperbolic plane. Given a curve I wonder how the tangent to a curve is defined in a plane with constant positive curvature. I ...
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50 views

smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ ...
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express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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1answer
35 views

invertible immersion is a diffeomorphism

Show that an invertible immersion $\varphi$:N $\to$ M is a diffeomorphism. Give a counterexample to this statement if N does not have a countable basis. I think I can use the Constant Rank Theorem. ...
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1answer
38 views

Substitution of variables in Laplacian

Suppose we have a function $u\colon \mathbb{R}^n \to \mathbb{R}$. Let $x \in \mathbb{R}^n$ and let $x=cy$ for a given constant $c$. How do I write $\Delta_x u(x) = \Delta_x [u(cy)]$ in terms of ...
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39 views

Notation on an estimate of the sectional curvature.

In a paper on the Ricci flow i am currently reading (http://arxiv.org/abs/math/0612095) the following estimate occurs several times (for example Lemma 4.1 and 4.2); $$\operatorname{sec}(g_0) \geq ...
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118 views

How do i show that dy/dx does not equals to zero

The question goes like this. Equation of a curve is $2x^2-3xy+y^2=5$ Find the equations of the tangent and normal to the curve at point $(4,3)$. Show that there is no point on the curve at which the ...