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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Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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41 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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74 views

Hamiltonian reduction in symplectic geometry

If $V$ is symplectic and $W^\perp \subseteq W\subseteq V$, then why is $W$ a pre-symplectic vector space? Why is $W/W^\perp$ symplectic?
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60 views

Path spaces and induced maps on tangent spaces

Let $M$ be a smooth manifold and $\Omega(M)$ the set of all piecewise smooth path in $M$. Let be $$ f: \Omega(M) \rightarrow \mathbb{R} $$ How can I define $$ f^*: T_{\omega}\Omega \rightarrow ...
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angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
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335 views

Berger's theorem on holonomy

Can someone clarify to me what the correct hypothesis of Berger's theorem are (if at all what I write is correct)? Theorem: assume $M$ is a Riemannian manifold, with irreducible reduced holonomy ...
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1answer
87 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
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408 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...
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105 views

conformally euclidean metric on riemannian surface

We know that in euclidean space $\mathbb{R}^3$ the euclidean metric induces on a 2-dimensional surface a riemannian metric which can be brought into conformal form by means of a local change of ...
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247 views

local isometry for riemannian manifolds is not transitive

Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds of the same dimension, and let $\phi: M_1 \to M_2$ be a smooth map. We say that $\phi$ is a local isometry if $g_2 (\phi_* X, \phi_* Y ) = g_1 ...
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176 views

Differentiability of the distance function and cut locus

Let $ M $ be a complete Riemannian manifold and let $ d: M \rightarrow R $ be the distance function from a given point $ 0 \in M $. I want to prove that $ d $ is a smooth function on $ M -(C(p)\cup ...
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76 views

How does $\operatorname{Ric} \ge 0$ guarentee the Busemann function is regular in the splitting theorem?

Cheeger-Gromoll's famous splitting theorem says If $(M,g)$ contains a line and $\operatorname{Ric} \ge 0$. Then $(M,g)$ is isometric to a product. I want to know how does $\operatorname{Ric} \ge ...
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176 views

Dimension of isometry group of complete connected Riemannian manifold

Given an $n$-dimensional geodesically complete connected Riemannian manifold $M$, we want to prove that the dimension of its isometry group is $$\dim {\rm ISO}(M) \leq \frac{n(n+1)}2.$$ Does it ...
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1answer
53 views

Minimum ratio between surface and volume in a riemannian manifold

In an euclidean three - dimensional space the sphere is the geometric figure with the minimum ratio $R=\frac{S}{V}$ with $S=4\pi r^2$ and $V=\frac{4}{3}\pi r^3$, so we have: $$R=\frac{1}{3}r$$ where ...
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200 views

Eigenvalues of the laplacian on a compact manifold without boundary

Let $ M $ be a compact manifold WITHOUT boundary. It is clear that the first eigenvalue of the Laplace operator $ -\Delta $ is $ \lambda_0=0 $. Now we suppose that M has constant sectional curvature ...
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101 views

Do carmo problem. exe 6 section 8

Calculate the mean curvature and the sectional curvature of the umbilic hypersurface of the hyperspace. please introduce a book that calculate this. or show how i can calculate this. This is a part ...
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61 views

Totally geodesic immersions

Let $ x: M \rightarrow \overline{M} $ be a totally geodesic immersion, where $ M $ is a $ k- $ dimensional Riemannian manifold and $ \overline{M} $ is a $ n- $ dimensional Riemannian manifold. Is it ...
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Question in do Carmo's book Riemannian geometry section 7

I have a question. Please help me. Assume that $M$ is complete and noncompact, and let $p$ belong to $M$. Show that $M$ contains a ray starting from $p$. $M$ is a riemannian manifold. It is ...
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93 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
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3answers
175 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
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1answer
77 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
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signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
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289 views

A question about geodesics of hyperbolic space

Let $ H^n $ be the the upper half space of $ R^n $ endowed with the conformal metric $ g=\frac{1}{x_{n}^{2}}|dz|^2 $ ($ |dz|^2 $ is the standard metric of $ R^n $). This space is the classical ...
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115 views

Singularity models of the Ricci flow

I faced this sentence in my studies on Ricci flow: The Bryant soliton is a singularity model for the degenerate neckpinch. Q1: What is the definition and meaning of singularity model? Can one model ...
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61 views

Are any two smooth Cauchy surfaces of a globally hyperbolic manifold diffeomorphic?

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold. It is known that there exists a smooth Cauchy surface $\Sigma\subset M$ and that $M$ is diffeomorphic to $\mathbb{R}\times\Sigma$. I suspect ...
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1answer
175 views

first variation of area

Does anyone know any good references or notes for proving the first variation of area under inverse mean curvature flow? I know what the result should be, but I dont understand the proofs really, and ...
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1answer
203 views

left-invariant n-form and metric on a Lie group

These two questions are from my exam practice question sets , which are quite similar. I got some problem understanding and solving both of them . For (a) , I can only substite $dx\wedge dy\wedge ...
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1answer
243 views

Divergence and Levi-Civita connection

Let $M$ be a level set of a function in $\mathbb R^3$. Then the mean curvature of $M$ is given by the trace of the second fundamental form which is a divergence term involving the Levi-Civita ...
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1answer
138 views

Riemannian metric for surface of negative Euler characteristic

So, I want to equip a surface of negative Euler characteristic with a Riemannian metric of negative curvature. I know from the uniformization theorem, that a metric of constant curvature exists ...
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131 views

Connected Sum of Riemannian Manifolds

As topological spaces, one can take connected sums of riemannian manifolds. Is there a way to give a Riemannian metric to the connected sum of two Riemannian Manifolds? If so is there a way to take ...
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122 views

Hopf-Rinow Theorem for Riemannian Manifolds with Boundary

I am a little rusty on my Riemannian geometry. In addressing a problem in PDE's I came across a situation that I cannot reconcile with the Hopf-Rinow Theorem. If $\Omega \subset \mathbb{R}^n$ is a ...
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1answer
129 views

Exponential map and connection

Suppose you have a Riemannian manifold $(M,g)$ and a point $p\in M$ fixed. Let $v: s\mapsto v(s)$ be a curve in $T_pM$. Now consider the map $f(s):=\exp_p(v(s))$. Can one get an explicit formula for ...
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1answer
320 views

Hodge Star Operator and definition of divergence operator on riemannian manifold

From my lecture notes, on a general Riemannian manifold $(M, g)$, the divergence operator $\operatorname{div}: T^\infty(TM)\rightarrow C^\infty(M)$ is defined as $\operatorname{div}X := ...
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1answer
80 views

Minimizing an expression (over the integers)

In the context of Hurwitz groups and manifolds, one comes by what wikipedia defines as a "remarkable" fact, that $1-\dfrac1 a -\dfrac 1 b - \dfrac1 c > 0$ has a minimal value of $1/42$ if $a,b,c ...
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1answer
161 views

Index of twisted Dirac operator

I do not understand a step in the proof of the Lemma 11.4.1 in the book "Analytic K-Homology" by Higson, Roe. Let $S$ be a Dirac bundle over a closed manifold $M$ and $D$ the corresponding Dirac ...
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What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
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347 views

Why is $\langle \operatorname{grad} f, X\rangle_g$ independent of the metric on a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold and let $f \in C^{\infty}(M)$. Let $X$ be a smooth vector field on $M$. In smooth local coordinates $(x^i)$ on $M$, we can write $g = g_{ij} dx^i \otimes dx^j$ as ...
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67 views

In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms

Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms? Thanks for your time.
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1answer
128 views

Time has dimension $2$ with respect to the Ricci flow scaling

Terence Tao in his lecture notes on Ricci flow has written: If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the ...
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1answer
96 views

Uniformly quasi-isometric patches over classes of Riemannian manifolds

Suppose $(M^d,g)$ is a closed, connected Riemannian manifold. Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
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1answer
86 views

Are there more embeddings $U(2) \hookrightarrow SO(4)$?

It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$. My ...
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1answer
36 views

Space of embedded surfaces with a common point

Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
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What is the volume of Complex Projective Space with Fubini-Study Metric?

I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
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1answer
98 views

Using Gauss-Bonnet to prove that geodesics have at most one point of intersection

Given an oriented Riemannian manifold $(M,g)$ of dimension $2$, such that $M$ has negative Gaussian curvature everywhere and $M$ is diffeomorphic to $\mathbb R^2$, I'm looking for a way to show that ...
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1answer
105 views

Prove Green formula

Let $(M^n,g)$ be an oriented Riemannian manifold with boundary $\partial M$. The orientation on $Μ$ defines an orientation on $\partial M$. Locally, on the boundary, choose a positively oriented ...
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1answer
73 views

How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?

Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$? Where $M$ is a ...
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172 views

Exponential Map

this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood of $\Gamma$ such that a point ...
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1answer
60 views

structure of Riemannian manifold of isometries from C^n to C^m

Does anyone know a reference which gives the properties (geodesics, geodesic distance, etc) of the Riemannian manifold of isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, which map zero ...
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Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
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164 views

Is a Riemannian metric positive definite or positive semidefinite?

From Wikipedia The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space, But a Riemannian metric is ...