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What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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Showing that the two-sheeted cone is is not a regular surface

I was given the following problem in class: Show that the two-sheeted cone, with its vertex at the origin, that is, the set $$\{(x, y, z) \in \mathbb{R^3}\,|\, x^2 + y^2 - z^2 = 0\}\text{,}$$ ...
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Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ... 1answer 53 views Prove: If$\int_{\phi}\omega = \int_{\psi}\omega$whenever$\phi$and$\psi$have the same endpoints, then$\omega=df$This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that$\int_{\phi}\omega$only depends on the endpoints$\phi(a)$and$\phi(b)$where ... 1answer 45 views Line Integrals in$\mathbb{R}^n$and Differential 1-Forms I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form$\omega$, every value of$x\in \Omega$maps to a linear function$\omega _x$, ... 1answer 43 views Gaussian curvature of a parallel surface Question 11 section 3.5 in Do carmo part c.Let a surfae x have constant mean curvature equal to c does not equal 0 and consider the parallel surface to x at a distance 1/2c. Prove that this parallel ... 1answer 18 views Mean curvature an asymptotic directions I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two ... 1answer 36 views How to show left-invariant frame? I am working on the following problem and I'm stumped. What do I need to show? What does "left-invariant frame" mean? Consider$\mathbb{S}^3$as the unit sphere in$\mathbb{R}^4$with coordinates ... 2answers 37 views How to show the symplectic group is a submanifold of$GL(n,\mathbb{H})$? I am trying to show that the symplectic group$Sp(n) =\{A\in GL(n,\mathbb{H})\mid \overline{A}^TA=I\}$is a regular submanifold of$GL(n,\mathbb{H})$but I am stuck. Any help would be appreciated. 3answers 110 views Are truth tables a valid method to prove an iff statement? I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ... 3answers 121 views Manifolds and their dimension Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on$\mathbb R^3$. I have difficulty in ... 1answer 41 views 2 dimensional Laplace's equation in polar coordinates The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that$\operatorname{div}(\cdot)$in two dimensional vector field could be written as $$\nabla \cdot u ... 0answers 28 views Show an immersion is locally one to one using the inverse function theorem Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint? 0answers 45 views Geometry of Curves and Surfaces The following is a question regarding Elementary Geometry of Differentiable Curves. Venture a guess of the number of irregular values of t for the trochoid with h = 1, \lambda = \frac{m}{n}, ... 0answers 45 views Geodesics of the Hyperbolic Plane. Using the coordinates \alpha=\log \frac{1+r}{1-r} and \theta where (r,\theta) are the usual polar coordinates, show that the segment of the y axis between (0,0) and (0,r) where 0<r<1 ... 1answer 33 views locate critical points/values and show where function is a regular surface Let f(x,y,z) = (x + y + z - 1)^2. a. Locate the critical points and critical values of f. b. For what values of c is the set f(x,y,z) = c a regular surface? a. So, to locate critical points ... 2answers 54 views Solving a certain differential equation when assuming a surface of revolution is minimal The problem is the following: Consider the surface of revolution$$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$If \textbf{q} is minimal, then r(t) = a\cosh(t)+b\sinh(t) for a,b ... 1answer 117 views Asymptotic Curves and Lines of Curvature of Helicoid I have a question that asks me to find the asymptotic curves and lines of curvature of the helicoid given by: x = v Cos[u], y = v Sin[u], z = c*u, for some fixed real c. Can you show me how best to do ... 0answers 49 views Prove that the tangent space TpM is vector space . We have to prove that (T_p M , + , *) is vector space " + " : T_p M \times T_p M \to T_p M (X_p + Y_p)(f) = X_p(f)+Y_p(f) So we have to prove that (T_p M ,+) is abelian group : ,, + " is ... 1answer 66 views Problem about parallel curve- differential geometry Let \alpha (s) , s\in [0,L], be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve \beta(s)=\alpha (s) +\lambda n(s), where \lambda is a positive ... 1answer 92 views Using index notation to write d^2=0 in terms of a torsion free connection. Let (M,g) be a Riemannian manifold and let \omega be a 1-form on M. I want to rewrite d^2\omega=0 in terms of the Levi-Civita connection. I can show the following:$$d\omega(X,Y) = ... 2answers 51 views How to identify a curve Suppose that a curve$\mathbf\gamma$in$\mathbb R^3$has constant strictly positive curvature function$\mathbf\kappa(s)$, and constant non-zero torsion function$\mathbf\tau(s)$. Prove that the ... 1answer 35 views parameterized ellipse, error in proof of a theorem? A question from the book "Elementary Differential Geometry" from A Pressley Consider the ellipse$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$, where$p>q>0$The eccentricity of the ellipse is ... 0answers 61 views tubular neighborhoods - problem Consider$\alpha:[0,L]\to \mathbb R^3$a smooth regular simple closed curve, arc-length parametrized. Denote its trace by$\gamma$. Let$\epsilon >0$be a real number and$N_\epsilon (\alpha)$the ... 1answer 63 views An immersive map is locally left invertible Question: Suppose that$m < n$, that$U$is an open set in$\mathbb R^m$and that$f : U \rightarrow \mathbb R^n$is a$C^1$function that has rank$m$everywhere in$U$. Show that for every ... 1answer 55 views Meridians of surfaces of revolutions First off, I know there is another question asking the same thing, but that one was concerning where to start, whereas for this one, I'm almost complete, but I can't get something at the end to ... 1answer 57 views Calculate the integral of a 2 form I am trying to compute the integral $$\int\int_{S}\frac{1}{x}dy\wedge dz+\frac{1}{y}dz\wedge dx+\frac{1}{z}dx\wedge dy$$ over an ellipsoid given by $$... 0answers 26 views I need help with the derivation of the equation of a tangent line at a point on a curve, and the arc length parameter s. I'm having difficulty seeing how under the arc-length parameterization the equation of the tangent line$$\frac{X-x}{dx}=\frac{Y-y}{dy}=\frac{ Z-z}{dz}=u$$can be written as ... 1answer 24 views Connection form uniquely determined by linearly independent \theta_1,\theta_2? I'm working through a tutorial for a differential geometry class. The question is: Consider the structure equations for a map \bar x:\mathbb R^2\to\mathbb E^2. Suppose that \theta_1,\theta_2 are ... 1answer 41 views Question related to tangent space of U(n) at a matrix g\in U(n) I was working on a homework problem that involved showing that the map f:U(n)\rightarrow S^1,g\mapsto det(g) is a submersion (which is given here) And the following question emerged: Given g\in ... 1answer 42 views Showing that a map h:S^2\rightarrow \mathbb{R}^4 is an immersion The Problem Let h:S^2\rightarrow \mathbb{R}^4 be a smooth map of the form$$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$Show that h is an immersion for any a,b\in \mathbb{R},a,b\neq 0,ab<0. Attempt ... 1answer 120 views Curvature of a curve lying on a sphere? This is a sample question from a multivariate calculus class. Any insight would be appreciated. Suppose the curve \mathbf{r} = \mathbf{r}(s) is parametrized by a natural parameter and lies on the ... 0answers 52 views Proving a curve of a logarithimic spiral is 1/s cot(theta) A unit-speed plane curve \gamma has the property that its tangent vector t(s) makes a fixed angle \theta with \gamma(s) for all s. Show that: (i) If \theta = 0, then \gamma is part of a ... 1answer 53 views Finding the tangent plane of a point of a curve when using implicit differentiation I need to find the tangent plane of this surface:$$(z-1)^3=\sin(y^2)e^{xz}$$at the point (0, \sqrt \pi ,1) I find dz \over dx and dz \over dy$${dz \over dx}={-e^{xz}\sin(y^2)z \over ... 1answer 49 views Parallel Curve of Regular Plane Curves Let$\gamma$be a regular plane curve and let$\lambda$be a constant. The parallel curve$\gamma^\lambda$of$\gamma$is defined as $$\gamma^\lambda(t)=\gamma(t)+(\lambda)n_s(t),$$ where$n_s$is the ... 0answers 42 views Given the normal vector n(s), determines the curvature k(s) and the torsion Given the normal vector n(s) of a curve$\alpha$, with non zero torsion everywhere, determines the curvature k(s) and the torsion$\tau$(s) of$\alpha$. I am first trying to show the following which ... 1answer 42 views Line containing a vector equation In Do Carmo 1.5.1d, it asks: Show that, for the parametrized curve$\alpha(s) = (a$cos$\frac{s}{c}, a$sin$\frac{s}{c}, b\frac{s}{c})$, the lines containing the normal$n(s)$passing through ... 2answers 71 views Equation of a plane from cross product I'm working from Do Carmo, and I ran into another snag. More specifically, 1.4.5: Given points$p_1, p_2, p_3 \in \mathbb{R}^3$, show that the following expression gives the equation for the ... 1answer 35 views Finding limit of a parametrized curve Related to this question: Parametrized curve tangent to a line I'm working on Do Carmo 1.3.5c, which is: Given a parametrized curve$\alpha(t) = (\frac{3at}{1+t^3},\frac{3at^2}{1+t^3})$and the ... 1answer 74 views Does there exist a surface homemomorphic to a torus with positive Gaussian curvature? This is a problem from the my last exam in Differential Geometry II and I didn't solve it. I'm studying again, but without success. So I need help. Does there exist a surface$S \subset ...
Let $g: \mathbb{R}^2 \to \mathbb{R}^2$ , $g (x, y) = (x^2-y^2, y)$ be a differentiable map. Let $r$ the line passing through $(1, 0)$ parallel to the $y-$axis. Prove that $g^{-1}(r)$ is a ...
I'm working from Do Carmo's book Differential Geometry, and I was a bit confused about one question - 1.3.5 in particular. The question asks: Let $\alpha:(-1,\infty) \to \mathbb{R}^2$ be a ...