Tagged Questions

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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first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
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Parallel transport on a submanifold

Let $M$ be a Riemannian manifold and $N$ an embedded submanifold of $M$. Now I have a geodesic $c \colon (-a,a) \to N$ with $a>0$ and $c(0)=p$ in $N$, s.t. $c'(0)$ is orthogonal to $T_pN$. Is the ...
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angle sum for triangle on helicoid

Given the helicoid $$\boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
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For submersion and submetry,why can we lift a geodesic “horizontally” to a geodesic?

A map $\sigma:X\to Y$ between locally compact complete inner metric spaces is called a submetry if $\sigma(B_r(p))=B_r(\sigma(p))$ for all $r>0$ and $p\in X$. Why is that a geodesic in $Y$ can be ...
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Commutativity of two vector fields. Prooving that $\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$

If $X$ and $Y$ are smooth vector fields with flows $\phi^{X}$ and $\phi^{Y}$, then starting at some $p\in M$, if we flow with $X$ for time $\sqrt t$, then flow with $Y$ for time $\sqrt t$ and then ...
Theorem 1.7 of Schottenloher's "Mathematical Intro to CFT" says that every conformal Killing field on a connected open subset $M$ of $\mathbb{R}^{p,q}$ for $n=p+q>2$ is of the form $$X^{\mu}(x) = 2 ... 0answers 107 views On differential geometry in Hilbert spaces Suppose that H is a Hilbert space and M\subset H is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the normal vector is onto on the ... 0answers 261 views Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence? Let X be an n-dimensional differentiable manifold and p \in X . Let (U, h, V ) for X around p with coordinates (x_1 , . . . , x_n ) in V , and let v_i , i = 1, . . . , n , be the basis of T_{... 0answers 210 views Chern class of tautological line bundle I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let E be the tautological line bundle over projective ... 0answers 100 views An orbit of a group action and the implicit function theorem Suppose that a Lie group G acts smoothly on a manifold X. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ... 0answers 89 views Following a polyline along the surface of a polygon that is twisted I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ... 0answers 64 views Stoke's theorem application to curl theorem. I did. Please can you check it? Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept w=Pdy∧dz +Q dz∧dx + R dx∧dy \in Ω^2(M) dim(M)=2 M is the subset of \Bbb R^2... 0answers 211 views Show that the projection map is Orientation preserving iff n is even My question is that Orient the unit sphere S^n in \Bbb R^{n+1} as the boundary of the closed unit ball. Let U be the upper hemisphere U ={x∈S^n |x^{n+1} >0}. It is a coordinate chart on ... 0answers 52 views Trivialization of a path of tamed almost complex structures I am wondering if the following result is true: Let (V,\omega) be a symplectic vector space and \{J_t\}_{0\leq t\leq 1} a smooth path of almost complex structures on V which are tamed by \... 0answers 82 views Pole of differential Let E : y^2 = x^3 + ax + b be an elliptic curve over the field K, char K \ne 0. We know that the differential \omega = \frac{dx}{y} is holomorphic in infinity because we can write it as \... 0answers 90 views Curvature form projective spaces Let T\mathbb{C}P^n tangent bundle over \mathbb{C}P^n. We have an hermitian metric on T \mathbb{C}P^n defined as h=\frac{dzd\bar{z}}{(1+|z|^2)^2}. If we consider Levi-Civita connection we can ... 0answers 392 views Chern classes tangent bundle. I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over \mathbb{C}P^n (T\mathbb{C}P^n) and develop the theory beginning from this ... 0answers 140 views Cotangent bundle of a complex projective space How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold? 0answers 134 views “Product” bundle notation. Let \newcommand{\Spin}{\operatorname{Spin}}M and M' be two manifolds, equipped with a principal \Spin_n and \Spin_{n'} bundle called P and P', respectively. Then there is an induced ... 0answers 163 views Are geodesic flows on surfaces with negative curvature Anosov? I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let \phi_t:TM\rightarrow TM be a geodesic flow on a compact surface M of ... 0answers 80 views How to define local sections on associated bundles? On a G-principal bundle P it's easy since G has a preferred element, namely e (identity), but there is no such element in arbitrary suitable space F to construct local sections for the ... 0answers 119 views characterization of the differentiable functions over a regular surface Let S be a regular surface. And let f:S\to \mathbb R be a differentiable function It's not hard to prove that if  W is an open set of \mathbb R^3 such  V\subset S\subset W, and f:W\mathbb ... 0answers 292 views Torsion of a closed curve Why the total torsion of a closed curve on a sphere is zero? What is the meaning of total torsion? Is it mean that we should calculate the torsion from point A to point A (i.e. since the curve is ... 0answers 84 views Does the derivative of the derivative depend on a choice of connection? Let X,Y be smooth manifolds and consider the infinite-dimensional manifold$$ C^\infty(X,Y) $$of smooths maps f: X \to Y. Note that there is an infinite-dimensional vector bundle E over this ... 0answers 116 views Reason for defining Riemannian curvature tensor and torsion tensor in particular way I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ... 0answers 209 views A question on tangent plane (from Do Carmo) From 'Do carmo Differential Geometry of curves and surfaces' On page 89, #9. Show that the parametrized surface S given by$$ \text{x}(u,v)=(v\cos{u},v\sin{u},au) $$Compute its normal vector N(... 0answers 110 views Divergence and curl united? In my post, In 2D we can define$$div(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{r} dCcurl_z(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{v} dC$$Where C is a ... 0answers 48 views How can we calculate the basis for right invariant vector fields from basis left invariant vectr fields I want to calculate the right invariant vector fields from left invariant vector fields. The fact I am using is that for a driftless system \dot X= XA we have \dot Y= -AY where Y=inverse Y 0answers 175 views How to calculate the Gaussian curvature of a non-embedded surface I know how to calculate the Gaussian curvature of an embedded surface using first and second fundamental forms, but how does one calculate the curvature of a non-embedded surface like the hyperbolic ... 0answers 98 views smoothness structure on a set I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ... 0answers 72 views an example of a curve such that… Given differentiable functions k(s),\tau(s) with domain s\in (a,b), there exist a differentiable function \gamma:(a,b)\to \mathbb R^3 such that k,\tau are it's respectively curvature and ... 0answers 118 views A question from Hamilton's Ricci Flow book by bennett chow On page 3 of the book before exercise 1.2, has written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ... 0answers 61 views deRham cohomology of a manifold with covering space S^{n} Let \pi: S^{n}\rightarrow M, n>1 be a covering map, M being an orientable manifold. Show that H^{k}_{deR}(M)=0 for 1\leq k<n. I know how to do for H^{1}_{deR}, but my argument fails ... 0answers 329 views do Carmo: near isolated zeros, killing field tangent to geodesic spheres Exercise 3.5b of do Carmo's Riemannian Geometry asks the reader to prove that given a Killing field X on a manifold M, an isolated zero p of X, and a normal neighborhood U of p in which X... 0answers 111 views Schwarz Lemma in Differential Form Suppose w=f(z) is a conformal self map of \mathbb{D}. From Schwarz Pick Lemma we have |\frac{dw}{dz}|=\frac{1-|w|^2}{1-|z|^2}. Could any one explain me In differential form how this becomes \... 0answers 192 views Circle tangent bundle over S^{2} Let S_{r}^{2} be a sphere of radius r and let TS_{r}^{2} be its tangent bundle. If SS^{2} = \{(x,p)\in TS_{r}^{2}| |p|=1 \} be the circle tangent bundle of non zero radius . Then are there ... 0answers 57 views Dual connections, bracket If \nabla is a torsionfree connection and (\nabla_{X}J)Y=(\nabla_{Y}J)X, J- an almost complex structure, and \nabla_{X}^{*}Y:=J\nabla_{X}(JY) its dual connecion. Is it correct to conclude that [... 0answers 148 views prove that is not conformal map I have fun with this website. It is very helpful and great. I have a question in geometry. I try to solve it but it needs imagination to figure out the point on the surface. This question makes me ... 0answers 49 views Why is that quantity a constant? Help needed! What have I done wrong here? Given the metric$$ds^2 = dr^2+r^2d\theta^2$$And$$R_{ij}=\nabla_i \nabla_j f(r)$$where \nabla_i is a covariant derivative, and f=f(r) is a scalar ... 0answers 95 views Integral over a Funnel in Fermi coordinates Suppose we are in the Hyperbolic plane, defined as$$ \{w = u + iv \mid v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$I am given a funnel F. This object is isometric to a ... 0answers 191 views De Rham cohomology of \mathbb R^3 without lines and a circumference I am trying to calculate De Rham cohomology of the following spaces: X=\mathbb R^3\setminus r where r is a line; Y=\mathbb R^3\setminus (r \cup \gamma) where r is a line and \gamma is a ... 0answers 90 views The standard connection in a regular surface is symmetric The concrete setting: Let M\subset \mathbb{R}^3 be a regular surface. A vector field X in M is a differentiable function X: M\to \mathbb{R}^3 such that X(p)\in T_pM. Here we are taking the ... 0answers 78 views Difeomorphisms and boundary conditions So I asked on physics.stackexchange, but got no answer, so I'll try here: I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism ... 0answers 188 views Integration on manifold, pullback Define the maps F_\pm:B^2\rightarrow S^2, (x,y)\rightarrow(x,y,\pm\sqrt{1-x^2-y^2}). Prove that for any \omega\in\Omega^2(S^2):$$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega ...
Let $\gamma : I \to \mathbb{R}^2$ be a smooth immersed curve, i.e. $\mathcal{C}^\infty$ and $\frac{d\gamma}{dt}\neq 0$ on $I$. I have found the following formula for the curvature $\kappa$ of $\gamma$,...