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 ...

learn more… | top users | synonyms (1)

2
votes
0answers
64 views

Gradient flows of functionals on manifolds

I'm reading some literature of ricci flows and on my way through it I quickly stumbled upon the gradien flow - one of the geometric flows. After searching on other books I found rigor definition of ...
4
votes
1answer
160 views

Passive and active coordinate transformation on a topological manifold.

Let us suppose we have $m$-dimensional smooth topological manifold $M$. Let $(U,\varphi)$ and $(V,\psi)$ be two charts on the manifold and $U \cap V \neq \emptyset$. For a point $p \in U \cap V$, we ...
3
votes
0answers
88 views

If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
1
vote
0answers
117 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
0
votes
1answer
85 views

Smooth Embedding

I need help showing that the following smooth map is not a smooth embedding: $f:\mathbb{S}^1 \to \mathbb{R}$ defined by $f(z)= \operatorname{Re}(z)$. I know that this map is not a submersion because ...
0
votes
1answer
132 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
1
vote
0answers
43 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
3
votes
0answers
162 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
7
votes
1answer
141 views

Tensor fields and vector bundles

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
1
vote
0answers
59 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
2
votes
0answers
43 views

Compatible connection on the associated vector bundle

Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says ...
5
votes
1answer
202 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
0
votes
1answer
17 views

If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$ Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)? I don't know what the chart map should be...
1
vote
2answers
52 views

space of sections of homogenious spaces

Let $G/H$ be a homogeneous space and then for homogeneous line bundle $L$ of $G/H$ the space of sections can be written as functions related to character of $H$. what about $\Gamma (G/H, L^2)$. then ...
3
votes
1answer
92 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
1
vote
0answers
59 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
1
vote
0answers
35 views

basis theorem in holomorphic tangent space

I know that if $(x^1, \cdots, x^n)$ is a local coordinate system in a manifold $M$ then $\{\frac{\partial}{\partial x^1},\cdots, \frac{\partial}{\partial x^n}\}$ forms a basis for the tangent space ...
8
votes
1answer
923 views

Applications of Differential Geometry in Artificial Intelligence

I am new to this wonderful site. I searched around a bit but I couldn't find any well-discussed posts on applications of differential geometry to artificial intelligence, or more generally to computer ...
0
votes
1answer
136 views
0
votes
1answer
765 views

Asymptotic Curves and Lines of Curvature of Helicoid

I have a question that asks me to find the asymptotic curves and lines of curvature of the helicoid given by: x = v Cos[u], y = v Sin[u], z = c*u, for some fixed real c. Can you show me how best to do ...
3
votes
0answers
30 views

A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a ...
2
votes
1answer
58 views

How can I prove the tangential acceleration equation?

This is my assignment for this weekend. $$a=a_{TT}+a_{NN} = \frac{d^2 s}{dt^2}\vec{T} + \kappa \frac{ds}{dt} 2\vec{N}$$ Actually, I want to why $a_N=\kappa (ds/dt)$ hold Please help me.
2
votes
2answers
84 views

connection on tensor bundle

I know that if $E$ and $F$ are two vector bundle with connection $\nabla^E$ and $\nabla^F$, then it is natural to define tensor connection $\nabla = \nabla^E\otimes 1 + 1 \otimes \nabla^F$ on $E ...
1
vote
1answer
67 views

Regular values and manifolds with boundary

Question: Let $X^m$ and $Y^n$ differentiable manifolds. $f:X\rightarrow Y$ a differentiable map. Show that if $\partial X=\emptyset$, $y\in Reg(f)$ and $f^{-1}(y)\neq\emptyset$, then $y\not\in\partial ...
3
votes
1answer
86 views

About a curious cross-product/determinant identity

Whilst proving the fact that one definition of area for a domain inside a parameterisation of some surface embedded in $\mathbb{R}^3$ is well defined, my lecturer made a claim "by linear algebra" that ...
3
votes
2answers
217 views

Explanation for the integral of differential forms

In our course of differential geometry we defined the integral $\int_{U} \omega$ of a differential form $\omega=f dx_1\wedge \ldots \wedge dx_n: T^nU \rightarrow \mathbb R$ with $U\subseteq \mathbb ...
7
votes
1answer
154 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
0
votes
2answers
94 views

why it is a projective space?

The unite sphere bundle of $TS^2$ , the tangent bundle of the 2-sphere, is the real projective space, $RP^3$. I can not understand the reason of it. And why the complement of the unit disc bundle of ...
0
votes
1answer
99 views

The tangent bundle over a manifold is trivial iff the manifold is paralelizable

Why is the tangent bundle over a manifold trivial if and only if the manifold is parallelizable? What additional condition do we need to impose on a fiber bundle if we want it to be trivial exactly ...
4
votes
1answer
38 views

Derivations are determined by their values on linear functions

How are derivations of the $\mathbb R$ algebra of germs of differentiable real functions on a manifold completely determined by their values in germs of linear functions? Are derivations of more ...
3
votes
1answer
78 views

Define curvature and curvature of a circle

Question:(a) Define the curvature function $\kappa$ of a plane curve. The curvature of $\kappa$ of a plane curve is the amount of turning in the osculating plane. In other words it decribes the speed ...
0
votes
0answers
84 views

Tangent space of the tangent bundle

Let $M$ be a differential $k-$manifold in $\mathbb R^n$, $g:TM\rightarrow\mathbb R$ given by $g(x,v)=|v|^2$, where $|\cdot|$ is the usual norm in $\mathbb R^n$. Show that $Reg(g)=Reg(g_{\partial ...
2
votes
0answers
28 views

non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
2
votes
1answer
97 views

Tangent Space to a manifold

So, I have a manifold $M=\{\mathbf{x}:\mathbf{\Theta}\left(\mathbf{x}\right)=\mathbf{0}\}$. I can also write $M=\{\mathbf{x}:\mathbf{F}(\mathbf{x})=\mathbf{c}\}$. Both functions are differentiable. I ...
1
vote
1answer
39 views

Normal vector of $\Gamma \times \mathbb{R}^+$ where $\Gamma$ is compact hypersurface

Let $\Gamma$ be a smooth boundaryless hypersurface of dimension $n-1$ in $\mathbb{R}^n$. Define $Q=\Gamma \times \mathbb{R}^+$. What does a normal vector of $Q$ look like? Because I want to compute ...
3
votes
0answers
59 views

Integrability of almost complex structure

If we want to check an integrability of an almost complex structure in $R^{4}$ is it enough to take vectors $X=X^{1}\frac{\partial}{\partial x^{1}}$ and $Y=Y^{1}\frac{\partial}{\partial x^{1}}$ and ...
0
votes
1answer
88 views

doing research/project on spacetime curvature

So I recently undertook the daunting task of presenting a project on general relativity for a differential geometry course. Does anyone have any suggestions for topic or topics to narrow it down to? ...
2
votes
1answer
143 views

constant speed of curve,regular curve, and reparametrization by an arc length

Question: Show that the curve $\alpha(t)=(sint,t,-cost)$ has a constant speed. Is this curve regular curve? Then find a reparametrization of this curve by an arc lenth. The curve of $\alpha$ has a ...
3
votes
1answer
127 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
2
votes
1answer
67 views

Very basic questions on chain rules and product rules

I have serious gaps in maths and would like to ask some basic questions. I know there is the following chain rule for the first derivative: $$ Dh(x) = Dg(f(x))Df(x)\quad\quad (1) $$ where $h(x) = ...
1
vote
1answer
130 views

Autoparallel submanifolds and geodesics

I have the following question in differential geometry. Any help is greatly appreciated. Let $M$ be an autoparallel submanifold of a manifold $S$ with respect to a connection $\nabla$. Let $\gamma$ be ...
2
votes
0answers
85 views

Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...
2
votes
3answers
232 views

Is it possible a trivial fiber bundle with nonzero holonomy?

Let $P\rightarrow M$ be a principal bundle with structure group $G$. Suppose that the bundle is trivial $M\times G$; is it possible to have a nonzero holonomy along some closed trajectory on $M$ for ...
8
votes
2answers
456 views

Definition of the Lie coalgebra

I don't understand how the Lie coalgebra is defined. The literature is never really explicit in how it is constructed. So I was wondering if anybody could supply me with a simple example of how the ...
1
vote
1answer
49 views

Is there a name for the result that homotopy groups are insensitive to removing submanifolds of sufficiently large codimension?

First some motivation. Consider $\mathbb{R}^n-\{0\}$. This is simply connected iff $n > 2$, since it deformation retracts to $S^{n-1}$. If instead we consider $\mathbb{R}^n - L$ where $L$ is a ...
2
votes
1answer
146 views

Inverse Function Theorem for Manifolds with Boundary

In Lee SM it is written that the inverse function theorem can fail for manifolds with boundary.As hint it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\to\mathbb{R}^n$ ...
2
votes
3answers
169 views

Finding the Total Curvature of Plane Curves

I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by $$\gamma(t)=(\cos(t),\sin(nt)), 0\leq t\leq 2\pi$$for each ...
0
votes
1answer
55 views

Some trivial questions about Tangent Spaces

I'm studying submanifolds $M \subset \mathbb{R}^n$ and now I've got some questions about tangent spaces. First question: Let $\gamma:I\subset\mathbb{R} \rightarrow \mathbb{R}^n$ be a path, which ...
4
votes
1answer
170 views

Geodesic eqautions and length of a curve in geodesic coordinate system.

About geodesic coordinates: Let S be regular surface. $p\in S$ $\gamma$ be unit speed geodesic on $S$ with parameter $v$ and $\gamma (0)=p$ $\tilde \gamma^v$ be unit speed geodesic s.t. ...
0
votes
1answer
121 views

Formula for the torsion of a regular curve paraemeterized by arc length

Let $\alpha:\, I \to \mathbb{R}^3$ be a curve parametrized by arc length $s$, with curvature $k(s) \neq 0$, for all $s \in I$. Show that the torsion $\tau$ of $\alpha$ is given by: $$ ...