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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Riemannian metric on $\mathbb{R}^n\setminus \{0\}$ induced by $\mathbb{R}^n$ expressed in polar coordinates.

I want to show that the metric induced on $\mathbb{R}^n\setminus\{0\}$ by its inclusion in $\mathbb{R}^n$ can be expressed in polar coordinates in the form $g=dr^2+r^2dS^n$ where $r=|x|$ for ...
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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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Recovering parallel transport

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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24 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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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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20 views

Isometry of covering space [on hold]

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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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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20 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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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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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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19 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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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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44 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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61 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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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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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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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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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
79 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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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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38 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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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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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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1answer
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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46 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? ...
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45 views

Riemannian metric of $3$-sphere

I know this probably seems like a dumb question, I have parametrised part of the unit $3$-sphere with $(x,y,z)\to (x,y,z,(1-(x^2+y^2+z^2))^{\frac{1}{2}})$ and now I'm trying to calculate the ...
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1answer
29 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
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23 views

Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

Overview This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a ...
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35 views

Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
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34 views

Closed geodesic minimizing properties

Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say ...
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Closed geodesics on real projective space

We have the result that all closed geodesics on $S^n$ must be contained with the intersection of $S^n$ and a plane. Hence all length minimising closed geodesics are single points. If we equip ...
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Green's function for Laplace operator in a conformally flat metric?

Given the Laplace–Beltrami operator $\nabla^2$, does there exists a closed form for the greens function $G$ such that $\nabla_x^2G(x,y)=-\delta(x,y)$, and $$ \nabla_x^2\iiint_{y^3}G(x,y)f(y)dy^3=-f(x) ...
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2answers
39 views

Expressing a metric as a sum of (possibly) many squares

Given a Riemannian manifold $M$ whose metric $g$ has zero curvature, it is known that we can find local coordinates $x^i$ such that $$g=\sum_{i=1}^{\dim(M)}(dx^i)^2.$$ Conversely, if the curvature ...
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1answer
89 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
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1answer
98 views

What is the smallest Euclidean space in which one can embed a given curved space?

Given a $d$-dimensional curved space, how many dimensions are required to embed it? As an example think of a sphere's surface, which is a two-dimensional curved space that can be expressed in ...
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29 views

Mean value theorem on Riemannian manifold?

Is there some generalisation of the classical mean value theorem for real-valued functions on an interval $$|f(x)-f(y)| \leq |\nabla f(c)||x-y|$$ for some $c$ between $(x,y)$ to the case where $f:M ...