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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Transformation induced by a spherical mirror

This is at heart a mathematical problem, but is best motivated in physical terms. I'll introduce a very special case and move on to the general case later. Special case An object, taken for ...
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158 views

Special types of Sasaki manifolds

i have a question to special cases of Sasaki-manifolds. Let $(M, g, \xi, \eta, \Phi)$ a Sasaki-manifold. In special case maybe $M=S^{2m+1} \cong \mathbb{C}^{m+1}$. This is a Sasaki manifold. But what ...
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17 views

Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
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22 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
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33 views

Function as a combination of 1-forms on a Riemann surface

My question is quite simple, I hope it's not also stupid.. Consider $R$ a Riemann surface and $\omega_1$, $\omega_2$ two $(1,0)$-forms (i.e. holomorphic forms) and $\varphi_1$, $\varphi_2$ two $(0,1)$-...
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35 views

Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
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25 views

Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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16 views

Unique Solution to Equation in Two Variables & Possible Use of the Implicit Function Theorem

Let $g(x) : R \to R$ be a continuous function; Consider the equation $ T(x,y) = y^3 -y^2 +(1+x^2)y - g(x)$ Show that for a given $x$ there exist a unique solution $y$ to the equation $T(x,y) = 0$. ...
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21 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
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20 views

Planar intersections of constant Gauss curvature K surfaces

Have they been studied? It appears they did not generate enough interest except the conic sections $K=0$. Do they give rise to curves of fourth order?
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20 views

I want to show that $ \mid D pr(m) \mid \leq \dfrac{1}{1-\Vert pr(m)-m \Vert \Vert h_{pr(m)}\Vert}.$

Let M be a smooth surface, and $U$ a neighborhood where the orthogonal projection pr is well defined. I would like to show that $ \forall m \in U$, if $ m \in S$, pr is differentiable on m and we ...
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58 views

Cauchy Sequence of Differentials and Point-Wise Limits

Let $D\subseteq R^2$ be an open and connected subset, and $\{f_{n} | D\to R^2\}$ a sequence of differentiable functions. Suppose that $\{(Df_{n}) | D\to Hom(R^2,R^2)\}$, the sequence of Jacobians, is ...
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33 views

Integration on manifolds with boundary

How can I define integral on manifolds with boundary? To use unity partition don't have I to deal with open sets of the same type, I mean, how can I be sure that there is a unity partition on my ...
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14 views

Equation for plugging in right-invariant vector fields in canonical connection?

Consider a matrix Lie group $G$ with Lie algebra $\frak g$ identified with left-invariant vector fields $\mathcal L(G)$. The $0$-connection is given by: $$ \nabla_{X^l}{Y^l}=\frac{1}{2}[X^l,Y^l]=\frac{...
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24 views

Tangent and normal vectors for parameterization of a straight line.

Hi everyone just a simple question. Suppose the curve $$r(t) = (x_0, t)\\ 0 \leq t \leq 1 $$ $x_0$ is some real number. We see that the unit tangent $T(t)= (0,1) = \dfrac{r'(t)}{\|r'(t)\|} $ ...
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66 views

Can one define 'geodesic' solely in terms of the betweenness relations among the points on that geodesic?

In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the ...
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35 views

Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
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52 views

The tension field is a vector field

Recall that the tension field of a function $f:(M,g)\rightarrow (N,G)$ is given in local coordinates by \begin{align*} & \Delta_gf^k+g^{ij}\hat{\Gamma}^k_{mn}\partial_if ^m\partial_jf^n \\ \end{...
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19 views

Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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44 views

How can we visualize a tensor?

I would greatly appreciate it if someone could explain to me how to visualize a tensor in the analogous way that we visualize a vector. Thanks!
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34 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
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86 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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54 views

Cohomology of the Riemann sphere

Let us note $\overline{\mathbb{C}}$ the Alexandroff compactification of $\mathbb{C}$ (i.e. the Riemann Sphere). I can prove that $$H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \simeq \...
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36 views

Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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51 views

Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
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26 views

Gradient of a function in different coordinates

Let $U\subset \mathbb{R}^3$ be an open subset endowed with a triple orthogonal coordinate system ($x^1,x^2,x^3$) and $f\in \mathcal{C}^\infty (U)$ be a smooth function. The vector field $\nabla f$ ...
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14 views

Extrinsic curvature for cylinder

Suppose we have the following metric describing a cylinder: $$ds^2=ud\rho^2+\rho^2d\phi^2$$ where $u$ is a function of $\rho$. We know the definition of the extrinsic curvature that is, $$K_{ij}=-\...
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43 views

local isometry between $x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v)$ and a helicoid

Define: \begin{equation} x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v) \end{equation} I want to show that there is a local isometry between $x(\mathbb{R}^2)$ and a helicoid: \begin{equation} y(u_{...
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26 views

For a minimal surface $M$ under Mean Curvature Flow, can it evolve between minimal surfaces continually?

I don't know much about this subject at all; I'm only just getting into it. As it turns out, a physicist friend of mine asked me a formulation of the following: Suppose $M$ is a surface in $\mathbb{R}...
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41 views

Book on differential Geometry with application to General Relativity

Does anybody know of a good book on differential geometry that has applications to general relativity and also focuses on geometrical intuition? I need a book that is not as rigorous as one that is ...
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32 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
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47 views

Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
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27 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
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18 views

Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = (0,0,0)^...
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21 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
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16 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations \...
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25 views

Apply flow of $V$ to a segment of a curve, Do you get covariant derivative?

Apply flow of $V$ to a segment of a curve, look at the velocity of the resulting new curve, a perturbation of the original curve. How is the new velocity (tangent) related to the original one (right ...
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37 views

Riemannian metric as an operator

In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - ...
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31 views

1-form that compute the area of parallelogram

Find a form on $\mathbb{R}^4$ that compute the area of parallelogram generate by any pair of vectors $\vec{a}, \vec{b}$ which are in the plane $\pi=\{\vec{x}\in\mathbb{R}^4| \vec{x}=\vec{p}+s\vec{u}+t\...
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27 views

real analytic functions on manifold

Let $M$ be a real analytic manifold of dimension $k$. Is it then always possible to find real analytic functions $f_1, \dots, f_k \colon M \to \mathbb{R}$, such that they are functionally independent ...
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33 views

On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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22 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
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55 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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33 views

geodesic flow is proper action

Good evening to everyone. I'm having a problem in the following setting: If I'm having a homogeneous manifold $M=G/K$, where $K \subset G$ is a closed subgroup, I can always find a $G$-invariant ...
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17 views

Integral curves of time dependent derivations

Question: Given smooth manifold $M$, with algebra of smooth functions deoted by $C(M)$ let $D_t$ be a time-dependent derivation of $C(M).$ Let $\hat{D}$ be a derivation of $C(M\times \mathbb{R})$ ...
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33 views

Defining a differential for quotients

Let $f \colon M \to N$ be a smooth map between smooth manifolds and $f$ being a surjective submersion. Assuming we have a proper Lie-group action $G$ on $M$, with only one orbit type and $G$ acts on $...
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46 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
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33 views

Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = [0,...
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25 views

On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

Let $M$ be a parallelizable manifold. Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$ for all $i,j$ ? If the answer is no, what kind of obstruction there is to find such a frame ? ...
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35 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...