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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Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
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Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
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Why are algebraic cycles rational?

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
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Find $\nabla_{\gamma'(t)}\gamma'(t)$. Let $S : z = x^2+ y^2$ be a surface in $\mathbb{R}^3$ with the induced metric and let $\gamma(t)$ be

Let $S : z = x^2+ y^2$ be a surface in $\mathbb{R}^3$ with the induced metric and let $\gamma(t)$ be a curve on $S$ given by $\gamma(t) = (t, t, 2t^2)$. For the arc length s$(t) = ...
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Forming a (1,1)-tensor field from a (0,2)- and (2,0)-tensor field

Let $A$ and $B$ be 2-tensor fields on a manifold, contravariant and covariant respectively. Prove that there exists a smooth (1,1)-tensor field $C$ with components defined by $$C^i_j = ...
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Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
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38 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
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12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
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38 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
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30 views

Sp(2n) as manifold

How to prove that $Sp(2n)$ is a manifold? We know that $Sp(2n)\subset Gl(2n)$ and $Gl(2n)$ is a manifold. Furthermore $Sp(2n)$ can be described as zeros of $A\mapsto A^TJA-J $, where $J$ is a ...
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48 views

Stokes Theorem on a sphere problem

I'm looking through my multivariable calculus notes and have come across a question I'm not sure I fully understand. It reads, "If $\omega$ is a differential form on $\mathbb{R}^3$ and $M$ is a sphere ...
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70 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
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16 views

Conical surface with negative curvature

I was reading some physics papers and I read about cones possessing negative curvature on the tip (and k = 0 everywhere else). Basically, to build these surfaces instead of removing a sector of the ...
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39 views

what is smooth embedding

I know definition but I don't use this definiton to solve question for example; $f : R → R^3$ given by $f(t) = (cos(2πt),sin(2πt), t)$ I showed that this is an injective immersion but this is a ...
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50 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
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19 views

Find $d\left(\frac{\partial\left(x,y\right)}{\partial\left(\delta_1,\delta_2\right)}\right)$ with the exterior product

Let $J_{\delta_1,\delta_2}^{x,y}$ denote the Jacobian $\partial\left(x,y\right)/\partial\left(\delta_1,\delta_2\right)$. Suppose I wanted to find $d\left(J_{\delta_1,\delta_2}^{x,y}\right)$ ...
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37 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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75 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in ...
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34 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
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32 views

Osculating Circle in Differential Calculus

I am working with an osculating circle as the curve of closest contact to a curve in differential calculus and my book takes some confusing steps that I do not understand. It says: Let $f(x)$ be the ...
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22 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
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28 views

Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
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36 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
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24 views

what does it mean that “not proportional”?

I'm reading this article. And here it says, "Let r(t) be a curve in Euclidean space, representing the position vector of the particle as a function of time. The Frenet–Serret formulas apply to curves ...
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28 views

Heat semigroup inequality

Let $M$ be a complete Riemannian manifold and $f \in L^p(M)$. I was wondering if there is any relation between $e^{t\Delta} u^p$ and $(e^{t\Delta}u)^p$, that is, if one is always less or greater than ...
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28 views

Integration of a 1-form over a “split curve”.

Bit of a strange question I can't really get my head around so apologies if it is ill-posed. Suppose we take a closed curve $\gamma: S^1 \to C \subset M$ in a Riemannian manifold $M$ and integrate ...
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51 views

Prove the directional derivative is a linear map

$\textbf{Directional Derivative:}$ Suppose that f is a smooth map of an open set in $\textbf{R}^n$ into $\textbf{R}^m$ and x is any point in its domain. Then for any vector ...
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$\phi : S \rightarrow \bar{S}$ an isometry, and $X:U \rightarrow S$ is a param. at a point p in S. Prove $\bar{X} = \phi \circ X$ is …

$\phi : S \rightarrow \bar{S}$ an isometry, and $X:U \rightarrow S$ is a param. at a point p in S. Prove $\bar{X} = \phi \circ X$ is a parametrization at $\phi(p)$ and$ E = \bar{E}, F = \bar{F}, G = ...
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20 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
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Sections of associated bundles isomorphism between spaces

I am reading some lecture notes which can be found here . They say that sections of $P\times_G F$ are represented by the functions $f:P\rightarrow F$ satisfying $f(pg)=\rho(g^{-1})\circ f$. Or ...
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Maximal circles and sphere.

Let $S^2$ the unit sphere given by the equation $x^2+y^2+z^2=1$. I have to prove that in every point $p \in S^2$ there is a maximum circle tangent in every direction to the plane $T_p S^2$. How can I ...
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65 views

Lie algebra of a lie group

In proposition 5.2 of Bump's Lie Groups, he states: Let $G$ be a closed Lie subgroup of $GL(n, \mathbb{C})$, and let $X \in Mat_n (\mathbb{C})$. Then the path $t \to exp(tX)$ is tangent to the ...
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53 views

Angle formed by parallel transporting a vector along two meridians of the sphere

Consider two meridians of a sphere. Let's call them $C_1$ and $C_2.$ Suppose they intersect at two points, $p_1$ and $p_2$, and form an angle of $\varphi$ between tangents of $C_1$ and $C_2$ at $p_1$. ...
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44 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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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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30 views

Is an hypersurface uniquely determined by an equation?

Consider $r:\Omega\to\Bbb R$ suffiently regular, $\Omega\subseteq\Bbb C^n$, $z_0\in\Omega$ s.t. $r(z_0)=0$. Then $r=0$ defines an hypersurface locally around $0$. My question is: is $r$ unique? By ...
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24 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 ...
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26 views

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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20 views

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 ...
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Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
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Proving path of motion is a Geodesic in general reletivity.

I am studying the work of Miguel Alcubierre, in particular his warp drive metric. A consequence of his metric is that the ship will travel on a geodesic and this is what I am trying to prove. I ...
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34 views

Conformal group, Minkowski metric

The conformal group is the subgroup of coordinate transformations that leave the metric invariant up to a scale factor. So under a transformation $x\rightarrow x'$ we have $g_{\mu\nu}(x)\rightarrow ...
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25 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
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59 views

Manifolds with smooth structure

One of the remark in my lecture notes said: In dimension $\leq 3$, every topological manifold has a unique smooth structure (up to diffeomorphism.) I don't quite understand what is a structure ...
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52 views

$\mathbb S^3$ parallelizable $\Rightarrow$ “uniform” lattice in $\mathbb S^3$?

Does the fact that the three-sphere is parallelizable imply that one can construct a "uniform" lattice in S^3? By uniform I mean that (assume the number of lattice points is large) any small ...
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32 views

Topology of “line integral convergence” on the space of curves

Let $C^1(I,\mathbb{R}^n)$ be the space of $C^1$ curves. Give it the topology that satisfies that convergence of a sequence of curves $\gamma_n \to \gamma$ occurs iff these conditions hold: a. ...
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Finding curves with special constraints. How do i verify the answer?

Let $O$ be a fixed point on the plane. Find all curves $\gamma : \mathbb{R} \to \mathbb{R}^2$ such that if $C(t_0)$ is the centre of the curvature of $ \gamma$ at $t_0$ then the angle $\widehat{ ...
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21 views

integration of differences of two inverse functions of CDF

When F, G are both cumulative distribution functions, $\int_0^1 | F^{-1}(y) - G^{-1}(y)|dy = \int_{-\infty}^\infty |F(x) - G(x)|dx.$ The paper (Vallender 74') says that "The equation follows from ...
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25 views

Prove using Math (Multivariable calc. or Diff. geometry proof) that $(x_u, y_u, z_u)\times(x_v; y_v; z_v)$ for a parametrization $X$ of $S$.

For a parametric surface $x = (x(u, v); y(u, v); z(u, v));$ the derivatives $x_u$ and $x_v$ are vectors in the tangent plane. Thus, their cross product = $(x_u, y_u, z_u)\times(x_v; y_v; z_v)$ is ...
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51 views

show that if $M \times \mathbb{R}^{n}$ is orientable than so is $M$

I need to show if $M \times \mathbb{R}^{n}$ is orientable than so is $M$, where $M$ is connected manifold. $R^{n}$ has standard orientation (determined by standard basis ) and by the assumption $M ...