# Tagged Questions

A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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### Covariant derivative of a constant inner product and how it decomposes into local coordinates

I wanted to confirm that my explicit (symbolic) computations of the covariant derivative of a constant inner product is (carefully) done right and correct. For the physicists (includes me), this is ...
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### Showing that the rank of the complex projective space is 1

I was assigned the task of calculating the rank of the complex projective space $\mathbb C P^n=SU(n)/S(U(1)\times U(n-1))$ and am not sure how best to approach that task. (looking in the ...
Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$\sum_kf_k^2=0\qquad (1)\qquad \&\qquad \sum_k|f_k|^2\not=0\... 0answers 43 views ### Normal coordinates I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism exp:U \subset T_pM \rightarrow V \subset ... 0answers 152 views ### Homogeneous metric on a homogeneous space G/K - is this the same as a G - invariant metric? I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold (M,g) on which a compact Lie group G acts transitively by isometries (this ... 1answer 172 views ### Are there spaces that 'look the same' at every point, but are not homogeneous? A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space '... 1answer 76 views ### Can a 1d space never be curved? I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature -1. I believe ... 1answer 61 views ### Does a homogeneous metrizable space admit a compatible homogeneous metric? Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ... 1answer 139 views ### Jacobi field along every geodesic? I stumbled over the question: Given a manifold M. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the 0 vector field does ... 1answer 78 views ### Normal coordinates and the metric tensor I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ... 0answers 36 views ### Isometry algebra implication from Riemannian covering I really wish that, if \pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h}) is a Riemannian covering, then \mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g}), where \mathfrak{i}(M,\mathrm{g})... 1answer 40 views ### Relationship between euclidean metric in sphere of radius r and the unit sphere. I want to show g_r=r^2g_1 where g_1 is the (Riemannian) metric in the unit sphere induced by its inclusion in \mathbb{R}^n and g_r is the metric in the sphere of radius r also induced by ... 1answer 33 views ### Gradient of Distant Function I am learning the Hessian comparison theorem on Riemannian manifold. It refers to the gradient of distant function. Fix x\in M. Let \rho(y)=d(y,x), and r:I\to M is a minimal geodesic curve with ... 1answer 33 views ### 'Large' closed subgroup I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold (M,h) admits a large closed subgroup K of the ... 1answer 391 views ### Review on Riemannian Geometry I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ... 0answers 41 views ### Example of locally symmetric spaces A locally symmetric manifold is a manifold with parallel curvature tensor \nabla R=0. Can you give an example except spheres, projective spaces and hyperbolic spaces? 1answer 30 views ### a tangent vector which does not fall on any geodesic Given a point on a manifold, is it possible that there is a tangent vector at that point which does not correspond to any local velocity of some geodesic? That is in that direction no geodesic exists ... 0answers 45 views ### Proving that \phi is orthogonal to the harmonic forms given \int\phi \;d\mathrm{vol}. I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold (M,g), the condition$$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$implies that \int \varphi ... 1answer 27 views ### Connection and curve Let \nabla be a connection on a Riemannian manifold and let the differential of a curve be given by$$c'(t)=c_1'(t)\partial_1 + c_2'(t) \partial_2.$$Now I was wondering how we define \nabla_{\... 0answers 109 views ### Singer & Thorpe Theorem on the curvature of 4-dimensional Einstein spaces I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by ... 0answers 143 views ### Degenerate subspace A null vector is a nonzero vector that is orthogonal to itself. If W is a subspace of V,let W^{\perp} = [v{\in} W : v{\perp}W]. W^{\perp} is a subspace of V called W perp. A subspace W of ... 1answer 25 views ### Locality of tensors part of definition? I am wondering whether linearity with respect to scalar functions f \in C^{\infty}(M, \mathbb{R}) is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ... 0answers 50 views ### Finding Riemannian metric from this geodesic In a d-dimensional Riemannian manifold, given a geodesic equation \gamma^i(t)=a^i\phi(tb^i),i\in 1\ldots d, where \phi:\mathbb{R}\rightarrow\mathbb{R} is an increasing function, a^i,b^i are ... 1answer 67 views ### Volume element and orientabality A volume element on an n-dimensional semi-Riemannian manifold M is a smooth n-form w such that w(e_1,\cdots, e_n) = \pm1 for every frame on M. How do I prove A semi-Riemannian ... 1answer 304 views ### Parallel Transport on a Cone Suppose we have a cone and we wish to parallel transport a vector w=(0,1,0) from along the curve \alpha(s)=(\sqrt{2}/2 \cos(v\sqrt{2}),\sqrt{2}/2 \sin(v\sqrt{2}),\sqrt{2}/2) from p=\alpha(0) to ... 1answer 62 views ### Ricci flow on surfaces : step in proof I am trying to realize the paper of richard hamilton's ricci flow on surfaces from the book of benett chow's Ricci flow : An Introduction.Here Hamilton denoted the trace free part of the Hessian of ... 1answer 27 views ### Proving the existence of certain vector field along a piecewise differentiable curve I am trying to understand proposition 2.5 in chapter 9 of do Carmo's "Riemannian Geometry". In the proof of the proposition he says: Let M be a Riemannian manifold and c:[0,a]\rightarrow M a ... 1answer 56 views ### Projection of surfaces in \mathbb{C}^2, \mathbb{C}P^2, \mathbb{C}H^2 to \mathbb{R}^3 As part of my thesis in Riemannian geometry, I study surfaces in \mathbb{C}^2, \mathbb{C}P^2 and \mathbb{C}H^2. Since visulation is always nice, I was wondering if there existed any "nice" ... 0answers 173 views ### Exercise 3.3 Riemannian Manifolds an Introduction to Curvature STATEMENT: Let \gamma(t)=(a(t),b(t)),t\in I(an open interval), be a smooth injective curve in the xz-plane, and suppose a(t)>0 and \dot{\gamma}(t)\neq 0 for all t\in I. Let M\subseteq \... 1answer 79 views ### Sheaf, étalé space with Riemann surfaces. Let f:X\rightarrow Y be an holomorphic map betwen two Riemann surfaces and let: \Gamma:={ (x,y)\in X\times Y|y=f(x) } \subset X\times Y be the graph of f. I have to show that (\Gamma,proj_{... 0answers 48 views ### Basic question: Riemannian Curvature is nondegenerate R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]} is the curvature with respect to the Levi-Civita connection \nabla of a metric g on a manifold M. Define the Riemann curvature ... 0answers 90 views ### Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics I think the answer to my question is known to many other people, but I'm still getting confused. Let G be a (possibly infinite dimensional also) Lie group and g be its Lie algebra. Consider the ... 1answer 91 views ### Riemann Roch Meromorphic section on a line bundle. Let g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*} defined by g(z,w)=(w^n z,\alpha w) where 0<|\alpha|<1. Let G be the cyclic group spanned by g and A the ... 1answer 139 views ### How to prove that the flat torus is indeed flat? The n-dimensional torus can be obtained as a quotient: T^n=\mathbb{R}^n/\mathbb{Z}^n. As pointed out here, the standard metric on \mathbb{R}^n is invariant under translation by the elements of ... 1answer 127 views ### Why are diffeomorphism-invariant PDE not elliptic? In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ... 2answers 132 views ### Curvature tensors and bivectors At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space \mathcal{R} of curvature tensors of the vector space V as the set ... 3answers 330 views ### What does it mean that we can diagonalize the metric tensor On a Riemannian manifold M, the matrix representation is diagonalisable, cause the tensor is symmetric. What is the physical meaning behind this? I mean, in Riemannian geometry, we always get a ... 1answer 28 views ### All riemannian isometries between open subsets of \mathbb{R}^n are affine I heard that there is a theorem of Liouville (Something like "Liouville's rigidity theorem") which states the following: Every Riemannian isometry between open subset of \mathbb{R}^n is affine. ... 0answers 43 views ### When a given family of curves are geodesics of some affine connection? Let M be a two-dimensional manifold and let \mathcal C be a family of smooth paths on M. How to understand whether this family is actually a family of (possibly reparametrized) geodesics of some ... 1answer 57 views ### Find a surface that has positive constant curvature that is not open subset of sphere Can some one find a surface that has positive constant curvature that is not open subset of sphere. I know every connected and compact surface with positive constant curvature is sphere. I need ... 0answers 28 views ### derivative of one parameter family of riemannian metrics Let X= \{ Riemannian \ metrics \ on\ M^n\}. Parametrized X as follows: define  f :[0,T) \subset \mathbb{R} \to X to be a surjective map. Now I want to define f'(t_0) for some  t_0 \in (0,T]... 0answers 33 views ### When are geodesically generated surfaces everywhere spacelike? Suppose that \langle M, g\rangle is a Lorentzian manifold, and that \xi is a timelike vector in T_pM, at some point p \in M. Let S be a surface consisting of points that lie on some ... 0answers 38 views ### Riemannian manifold and coordinate transformation Given a manifold \mathcal{M} with fixed "shape" (say a hemisphere), we may define two sets of Riemannian metrics and connections for \mathcal{M}, say g_{ij},\Gamma_{i,j}^k and g'_{ij}, \Gamma_{... 1answer 35 views ### Local expression of hermitian metric I have really hard times reading Zheng's Complex Differential Geometry and I find the following sentence especially baffling (sec. 7.4, page 170): "Let M^n be a complex manifold. A Hermitian metric ... 1answer 97 views ### Index notation. I have a very basic question concerning index notation, normally used in physic papers. I have never used index notation and it is very difficult to me to translate from index free notation, even in ... 0answers 35 views ### How does the Schrodinger's potential transformer if the metric conformally transformers? Given Schrodinger's equation$$ (-\nabla^2+V)\psi=E\psi  and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...