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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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 ...
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The Weierstrass-Enneper representation, the Gauss map

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 ...
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25 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 ...
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Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
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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 ...
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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 ...
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63 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 ...
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39 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 ...
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33 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 ...
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31 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 ...
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23 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 ...
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19 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 ...
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24 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 ...
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26 views

Recovering parallel transport [on hold]

Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p\in M$ and let $c:I\rightarrow M$ be an integral curve of $X$ thorough $p$ i.e $c(t_0)=p$ and ...
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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 ...
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87 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 ...
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22 views

Isometry of covering space [closed]

Let $M$ be a compact Riemannian manifold. Consider a covering space $N$ of $M$, with the pull-back metric from $\pi : N \to M$. Given a point $x \in M$, and a couple of points $y, z \in \pi^{-1}(x) ...
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25 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?
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24 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 a some geodesic? That is in that direction no geodesic ...
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28 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 ...
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22 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 ...
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28 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 ...
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20 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 ...
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20 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 ...
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28 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 ...
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45 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 ...
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62 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 ...
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33 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 ...
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1answer
18 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 ...
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36 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" ...
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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 ...
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62 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 ...
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33 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 ...
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35 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 ...
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1answer
70 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 ...
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30 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 ...
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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 ...
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20 views

Exponential map and convergence

Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ be a smooth function. I consider the expression $\exp_y^{-1}(x)(f)$: then it follows that it converges to ...
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56 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 ...
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3answers
86 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 ...
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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. ...
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29 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 ...
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1answer
40 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 ...
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11 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 ...
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23 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 ...
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20 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}, ...
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17 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 ...
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49 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 ...
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25 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 ...
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50 views

Diameter of the Grassmannian

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? ...