# 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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### 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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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 $\... 0answers 90 views ### Why are symplectic manifolds and Riemannian manifolds so different? They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ... 0answers 56 views ### Cohomology of the Riemann sphere Let us note$\overline{\mathbb{C}}$the Alexandroff compactification of$\mathbb{C}$(i.e. the Riemann Sphere). I can prove that $$H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \simeq \... 0answers 39 views ### Hyperbolic isometries and finite order elements I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of \text{Isom } H^n, the finite order elements ... 0answers 52 views ### Surjectivity of the exponential map on SO(2n)/U(n) Let M:=SO(2n)/U(n) the homogeneous space of all orthogonal almost-complex structures on \mathbb{R}^{2n}. When n=2, it is known that M is just the 2-sphere. 1) On the 2-sphere, the ... 0answers 26 views ### Gradient of a function in different coordinates Let U\subset \mathbb{R}^3 be an open subset endowed with a triple orthogonal coordinate system (x^1,x^2,x^3) and f\in \mathcal{C}^\infty (U) be a smooth function. The vector field \nabla f ... 0answers 44 views ### local isometry between x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v) and a helicoid Define: $$x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v)$$ I want to show that there is a local isometry between x(\mathbb{R}^2) and a helicoid: y(u_{... 0answers 28 views ### For a minimal surface M under Mean Curvature Flow, can it evolve between minimal surfaces continually? I don't know much about this subject at all; I'm only just getting into it. As it turns out, a physicist friend of mine asked me a formulation of the following: Suppose M is a surface in \mathbb{R}... 0answers 48 views ### Book on differential Geometry with application to General Relativity Does anybody know of a good book on differential geometry that has applications to general relativity and also focuses on geometrical intuition? I need a book that is not as rigorous as one that is ... 0answers 32 views ### Orientability of differantiable manifold of orthogonal matrices I want to find out if differentiable manifold of matrices M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9 is orientable. It is only worth proving that orthogonal ... 0answers 48 views ### Why say “exists smooth structure” instead of “is a smooth manifold”? As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ... 0answers 27 views ### When is a stable domain in a minimal surface area minimizing? A stable domain D in a minimal surface S\subset \mathbb{R}^3 is a domain for which the area-functional A(t):=\int_{S_t}dS_t has non-negative second derivative, i.e. A''(0)\geq 0, for all ... 0answers 18 views ### Finding braches of equilibria Consider the system of two equations in three variables (x,y,z):$$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$The point x^* = (0,0,0)^... 0answers 21 views ### Mean curvature submanifold Consider S^{N-1} the unit sphere and let us focus our attention on the cap$$ 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 ... 0answers 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 \... 0answers 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 ... 0answers 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) - ... 0answers 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\... 0answers 28 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 ... 0answers 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 ... 0answers 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 ... 0answers 34 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 ... 0answers 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}) ... 0answers 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 ... 0answers 49 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 ... 0answers 35 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,... 0answers 26 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 ? ... 0answers 37 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 ... 0answers 24 views ### Normal vector field for an immersion I know that a hypersurface$M$of a riemannian manifold$N$is orientable iff there exists a globally defined unit normal vector field$\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ... 0answers 28 views ### On Automorphism on Space of Smooth Functions Given a Operator$T$(an Automorphism) on the subspace$X$of Smooth functions on$\mathbb R ^n$,$\mathcal C^\infty(\mathbb R^n)$$$X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp \,(u)\... 0answers 9 views ### Metric of the doubling i.e circular billard table Let D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0) denote the doubling in \mathbb{R}^2 i.e. the metric space which one gets after gluing two closed balls of radius 1 with their induced metric coming ... 0answers 15 views ### parameterization of surface of revolution? parameterization of surface of revolution formed by revolving the x=\cosh z around z axis , i thought the it as$$x=\cosh z \cos \theta ,y=\cosh z \sin \theta ,z=z$$Hence the surface can be ... 0answers 19 views ### Showing that a mapping is an isometry Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution (x, y, z) = (\cos{\... 0answers 26 views ### How do I show that the reparametrization of a pre-geodesic is pre-geodesic? So this is probably a very silly question but I need to prove the reparametrization of a pre-geodesic is a pre-geodesic. The hint in my book says it's obvious, so I am clearly missing something. 0answers 34 views ### Proper patch in the differential geometry I have a question that coincides with this question. Proving that every patch in a surface M in R^3 is proper. Prove that if \mathbf{y}:E\to M is a proper patch, then \mathbf{y} carries ... 0answers 15 views ### Finding a zero of homogeneous parts of polynomials using zero of those polynomials Let f, g \in \mathbb{Q}[x_1, ..., x_n] be polynomials of degree d. Let F and G denote the degree d portions of f and g respectively. Suppose there exists a non-singular point \mathbf{... 0answers 25 views ### Proving the invariance of the mixed product under direct congruent transformations Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors:$$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle  \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle  \overrightarrow{... 0answers 30 views ### Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space? Grow a square in the hyperbolic plane until its vertex angles become$\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is$-1$, the sides of the resulting hyperbolic ... 0answers 45 views ### Discretizations of Differential, Geometric and Topological Notions I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ... 0answers 16 views ### Apply “thickness” to a minimal surface By definition a minimal surface has no volume. But my goal is to give the minimal surface some volume. Is there a mathematical way to do this? I found a thread in a forum of a math visualization tool ... 0answers 31 views ### The map$f:M\rightarrow N$is differentiable, where$M,N$are$m$-manifolds. Suppose the map$f:M\rightarrow N$is differentiable, where$M,N$are$m$-manifolds with$M$compact and$N$connected. If the degree of$f$is 1, then$f$is surjective? 0answers 62 views ### what does it mean to have inner product of$S^2$and$R^3$? It may be that the title of my question is wrong but i am writing this question because i am struck while reading this paper Brownian motion on rotational group Where$^*\mathscr{f} $is transpose of ... 0answers 48 views ### Zero gradient in$L^2(M)$I'd like to show that for$u \in L^2(M)$, for M a compact, connected Riemannian manifold, if$\nabla_g u = 0$(i.e$\forall XC^{\infty}$- vector field on$M$,$\int_M u \hspace{1mm} \text{div}_g X =...
This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...