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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Show that $-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|$

Here $\text{Rm}$ is the curvature tensor. When I try to compute $$-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|,$$ I compute ...
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23 views

Compute of a inequality about cutoff function

How to compute the inequality with red line in the below picture ? It seems to integrate the inequality 1 , but I don't know why there is $e^{CMt}$ and where the $\int_0^t\varphi|\nabla^2 \varphi|$ ? ...
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Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...
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How to define the smooth of vector field on Riemannian manifold? [closed]

$(M,g)$ is a Riemannian manifold, if $X$ is a vector field on $M$, I think for differential points $p,q\in M$, $X_p$ and $X_q$ are belong to differential space $T_pM$ and $T_qM$, I can't image how to ...
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30 views

Image of a path $\gamma$ is totally geodesic $\Rightarrow \gamma$ is a reparametrization of a geodesic?

Let $M$ be a Riemannian manifold. Assume $\gamma$ is a path in $M$ , such that it's image is a totally geodesic submanifold of $M$. I am trying to prove (the seemingly trivial result) that $\gamma$ ...
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32 views

Divergence formula

I would like to obtain the following equation : $\text{div}\left( \frac{\nabla u}{\sqrt{|\nabla u|^2 + \epsilon^2}} \right) =\frac{1}{\sqrt{\epsilon^2 + |\nabla u|^2}} \left( g^{ij} - \frac{\nabla^i ...
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41 views

Betti number of nonnegative Ricci curvature and positive scalar curvature closed 3-mfd

Suppose that $M^3$ is a closed 3-manifold with nonnegative Ricci curvature and positive scalar curvature, I think $b_1(M^3)\leq 1$. Is this right and is there a quick cut proof?
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33 views

Exchange of $\partial_t$ and inner product

If I'm right ,we define the inner product of tensor on Riemannian manifold as follow: $$ <T_{ij}^k,T_{ij}^k>=g_{il}g^{jm}g^{kp}T_{jk}^iT^l_{mp} $$ So ,if the metric $g_{ij}$ evolve under Ricci ...
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40 views

Curvature form and Riemann curvature

So I read that the curvature form and Riemann curvature are actually equal, but how do they show this result. $$\Omega(X,Y)=R(X,Y)$$ We note that the right hand side is the Riemann curvature tensor....
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27 views

Inverse metric coefficients in normal coordinates

I know that in normal coordinates centered at $p\in M$ the first order partial derivatives vanish at $p$, is the same thing true for the inverse metric coefficients, or is there a counterexample?
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51 views

Vertical lifts on vector bundles

So let us assume we have a vector bundle $\pi:E\rightarrow M$ and $\nabla$ a connection on $g$.I know that if $v\in E_p$, then there is the standard embedding of $v$ in $T_eE$ where $\pi(e)=p$. Namely ...
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29 views

Composite killing vector field

Let $f$ be a smooth function and $\nabla_V V = 0$. Prove that $fV \neq 0$ is a killing vector field, if and only if $f$ is a constant and $V$ is killing. If I assume that $f$ is constant and $V$ is ...
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2answers
67 views

Isotropic left invariant Riemannian metric on $GL_n^+$?

I am trying to see if it's possible to construct a left invariant isotropic Riemannian metric on $GL_n^+$. (the group of $n \times n$ invertible real matrices with positive determinant) (When by ...
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1answer
56 views

Where is the $\int_0^t\Delta F$?

$Rm$ is curvature tensor , and the $g_{ij}$ meet Ricci flow. For getting the inequality 1 in the below paper's 191th page , I compute that: $$ \because ~~~\frac{\partial F}{\partial t}\leq \Delta F+\...
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1answer
28 views

Curvature tensor in exponential coordinates

So I've been computing the Riemann curvature tensor in normal coordinates centered at the point $p\in M$, but I am getting that the curvature is zero at the point $p$. I don't think this is correct, ...
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1answer
16 views

Smooth local frame extends to a local chart

This is kind of a dumb question, but I was wondering if given a local orthonormal frame,$s_i$, in a Riemannian manifold, does it extend to a local chart $\varphi(p)=(x^1(p),...,x^n(p))$ such that $\...
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15 views

Reference for Riemannian geometry of Grassmanians?

Can someone recommend a reference for the Riemannian geometry of Grassmanians (for instance of $k$ planes in $R^n$)? I just think this would be a cool example to get to know well.
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24 views

Given two metrics $g,$ and $h$, related by $h = e^{\frac{2f}{n}}g$, what is the relation between $\star_g$ and $\star_g$?

Given two metrics $g,$ $h$, and a smooth function $f$ on a riemannian manifold $M^n$, supposing that $g$ and $h$ are related by $h = e^{\frac{2f}{n}}g$, what is the relation between $\star_g$ and $\...
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1answer
38 views

Counterexample to the form of Gromov compactness theorem without a Ricci curvature bound

Gromov compactness theorem states that in a class of Riemannian manifolds that have a uniformly bounded diameter and uniformly bounded below Ricci curvature every sequence of manifolds has a ...
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42 views

Show that $V$ is parallel iff $V'(t)$ is orthogonal to $T_\gamma(t)M$, where $V'(t)$ is the time derivative of $V$.

let $S$ be a surface in $R^3$ with induced Riemannian metric. let $\gamma : I ->S$ be a smooth curve on $S$ and $V$ a vector field tangent to $S$ along $\gamma$ . $V$ can be thought of as a ...
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37 views

Geodesics of the $\mathbb{S}^n$ are great circles

I am trying to show that the geodesics of $\mathbb{S}^n$ are the great circles, as an exercise for my introductory Riemannian geometry class. I don't really know how to go about this. I suppose that ...
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The covering manifold of a complete Riemannian manifold is complete

This question is from Lectures on the Geometry of Manifolds by Nicolaescu. (Exercise 6.2.8 b) Let $(M,g)$ be a complete (connected) Riemannian manifold and let $(\tilde{M},\tilde{g})$ be its ...
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45 views

Why does the halfplane model of the hyperbolic plane involve only the upper half of the plane?

The Hyperbolic metric $$ s= \pm \int \frac {\sqrt{1 +y'^2} \, dx}{y} $$ for geodesics in $ \mathbb H^2$ integrates to a full circle, but why only the upper half is considered? The query is about ...
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Exponential maps depends on Riemmanian Metric?

This may be a silly question but I am new to Riemmanian Geometry. If I have two different Riemmanian Metrics $g_1,g_2$ on a smooth manifold $M$, then do the geodesics on the Riemmanin Manifodls $&...
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Applications of the Hurwitz Theorem on Number of Automorphisms?

So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem: Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic ...
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49 views

Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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1answer
76 views

Lee's Riemannian Manifold, Zero curvature implies flatness

So I'm confused about a part of the proof Lee gives for zero curvature implies flatness, this is theorem 7.3. Proof:Let ${E_1,...,E_n}$ be an orthonormal basis at $p\in M$. Let $(x^i)$ be a ...
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39 views

Conformal metric

I'm trying to solve the following : Let $D$ be a simply connected region stricly included in $\mathbb{C}$. Let $\mathbb{D}$ be the open unit disk. Let $f \in \text{Hol}(D)$ be a bounded function ...
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71 views

Conformal immersions from surfaces into 3-manifolds

Let $f:(S,g) \to (M,h) $ be a smooth immersion of a compact surface into a 3 - manifold. Is it true that there exists a diffeomorphism $\phi: S \to S$, such that the metric $(f \circ \phi)^*(h)$ is ...
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How to show the inequation by using Hessian comparison.

In the below picture ,how to show the inequation 1? In fact,I'm not familiar with Hessian comparison.So, hope a detail answer , Thanks very much. The below picture is form 194th page of here ~~~~~~~~...
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40 views

doubling property on manifold

I am learning the Geometric Measure Theory, and curious about how to generalize the covering lemma on manifold. However I am stuck at the doubling property: Let $(X,d,\mu)$ be a metric space with ...
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1answer
83 views

Einstein manifolds and topology

Given a Riemannian manifold $(M,g)$ with Ricci tensor $ R_{mn} = k g_{mn} $. Suppose the Ricci scalar you get is $$ R > 0 $$ What can you tell about the manifold $globally$ ? In particular, can ...
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1answer
67 views

Show that $\frac{\partial\Gamma_{jk}^i}{\partial t}=\nabla \rm Rm$

There is a explain on my book,as the below picture. $\rm Rm$ is curvature tensor, in fact,I don't know why the curvature tensor be denoted by $\rm Rm$, I think it's should be $R_{ijkl}$. Besides $\...
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1answer
59 views

Comparing between intrinsic and external metrics on submanifolds of Riemannian manifolds

$\newcommand{\til}{\tilde}$ Let $(M,g)$ be a Riemannain manifold. Denote the induced Riemmann distance function by $d^M$. Let $S \subset M$ be an embedded connected submanifold. We have two natural ...
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Definition of geodesic closure

What is the definition of the "geodesic closure" of an open subset $U\subsetneqq M$?, where $M$ is a compact riemannian manifold with boundary.
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1answer
26 views

Book request-Riemannian geometry and control theory

I am looking for a book dealing with Riemannian geometry with a control point of view. Thanks
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68 views

Second fundamental form and metrics

Suppose $M$ and $N$ are orientable manifolds, $f: M \to N$ is a smooth embedding and $g$ is a Riemannian metric on $N$. When $M$ has codimension $1$ and $\vec{n}$ is a prefered unit normal section of $...
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23 views

Directional derivative near the bounder .

Let $\varphi:B^n\rightarrow B^n$ be a differential homeomorphism, $B^n$ is unit ball of $R^n$ , and $\overrightarrow{n}$ is outer normal vector of $B^n$. I feel that $$ \frac{\partial \varphi}{\...
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1answer
86 views

Geodesic Lines on Covering Maps

So I'm not sure how deck transformations work into this problem. I've established the following so far. Let $\pi:\tilde{M}\rightarrow M$ be the universal covering map. We may suppose that $M$ is ...
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1answer
60 views

Inequations about curvature tensor.

How to get two inequations 1 and 2? At the 1, I don't know how to use the Cauchy inequation . At the 2, I absolutely start from where.. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
37 views

Does the set of null vectors of an indefinite scalar product determine the product up to scale?

If $V$ is a vector space with index 1, let $g$ and $\hat{g}$ be scalar products, and denote \begin{align}\Lambda &= \{v \in V \mid g(v,v)=0\} \\ \hat{\Lambda} &=\{v \in V \mid \hat{g}(v,v)=0\}...
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85 views

different definition of connection on bundle

I have encountered two definitions of connection in two different setting namely in vector bundle and principal bundle and I don't see the equivalence in these two setting. In vector bundle setting: ...
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102 views

Diagonalizable operators on compact Riemannian manifolds

Let $M$ be a compact Riemanniam manifold and $\Delta$ the Laplace-Beltrami operator. Let $\lambda_0 \leq \ldots \leq \lambda_j \leq \ldots$ be the eigenvalues and $\phi_0, \ldots, \phi_j, \ldots$ be ...
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Two inequations of curvature tensor.

How to get the two inequations 1 and 2. And I think the RHS of 1.4.2 should be $$ \Delta(\nabla R_m)+2R_m*(\nabla R_m) $$ Besides, What's the define of $A*B$? Soryy for my lazy, the below ...
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Why $-g^{ij}(\nabla_jR_{kl}+\nabla_kR_{jl}-\nabla_lR_{jk})=\nabla Rm$

Why $-g^{ij}(\nabla_jR_{kl}+\nabla_kR_{jl}-\nabla_lR_{jk})=\nabla Rm$? Whether the $\nabla Rm(X,Y,Z,W,T)=\nabla_XRm(Y,Z,W,T)$?
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Why a 2-form $\varphi$ can be regarded as an element of the Lie algebra $so(n)$?

Why a 2-form $\varphi$ can be regarded as an element of the Lie algebra $so(n)$?Whether the $so(n)$ is special orthonormal space ?
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113 views

Direct proof of the second Bianchi identity

Let $X$,$Y$,$Z$,$W$ be vector fields on a riemannian manifold, and let $R(X,Y)W$ be the riemannian curvature: $$ R(X,Y)W = \nabla_{X}\nabla_{Y}W - \nabla_{Y}\nabla_{X}W - \nabla_{[X,Y]}W $$ Let $g$ ...
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1answer
38 views

Question about the metric tensor

From wikipedia: The metric can be written in the form $g=g_{ij}dx^i \otimes dx^j$. The metric is thus a linear combination of tensor products of one form gradients of coordinates. If we denote the ...
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1answer
19 views

Connection under the frame change.

In the below picture ,I don't know how to compute the red line 2. But if accept the red line 2, I can compute the red line 3. But there still be a little question, that at last ,I have $$ \Gamma_{ib}...
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1answer
24 views

Why the frame remain orthonormal under the flow?

Why the frame will remain orthonormal under flow ? At the beginning ($t=0$),if we choice a good frame that $g_{ij}=\delta_{ij}$,it is obvious that ${F_a}$ is orthonormal. But under the flow , $F_a(t)$...