# 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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### Proportinal line elements imply preservations of angles.

Consider a Riemannian manifold $(M,g)$, and a variation of the line element $\delta ds^2$ that is proportional to the original line element $ds^2$. This is $\delta ds^2=c ds^2$ for some constant $c$. ...
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### First and second fundamental form with rotational surfaces (check)

I'm working out some examples for surfaces in differential geometry. I was working out simple rotational surface, but I think I've done something wrong. Let $\gamma\left(t\right)$ a curve ...
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### Product-like metric on a pseudo-Riemmanian manifold foliated by Lie group orbits

Suppose we have an $n$-dimensional pseudo-Riemmanian manifold $(M,g)$ on which a connected Lie group $G$ acts isometrically (I am most interested in the Lorentzian case if it matters). Suppose that ...
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### Conformal class of $\mathbb S^n$ [on hold]

What can we say about the conformal class of the sphere $\mathbb S^n$?
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### Total Gaussian curvature of a flat surface with cone singularities

Let $S$ be a surface and $g$ a riemannian metric on $S$ which is flat with finite number of isolated conical singularities of cone angle $\theta_i>2\pi$. I have two questions: 1) of course the ...
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### Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
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### About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
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### Hyperbolic metric geodesically complete

Consider the upper half plane model of the hyperbolic space ($\mathbb{H}$ with the riemannian metric $g=\frac{dx^2+dy^2}{y^2}$). It is known that $(\mathbb{H},g)$ is geodesically complete, which means ...
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### “Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
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### Uniqueness of minimizing geodesic $\Rightarrow$ uniqueness of connecting geodesic?

Let $M$ be a complete connected Riemannian manifold. Fix $p \in M$. Assume every point in $M$ has a unique minimizing geodesic connecting it to $p$. Is it true that for every point, the only ...
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### Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann ...
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### How do I find the induced Riemannian metric of a real smooth complete intersection?

If I have a smooth complete intersection of $f_1,\ldots,f_k \in C^\infty(\mathbb{R}^n)$, presented as the vanishing locus $$f_1 = 0 \text{ } \cdots \text{ } f_k = 0$$ in $\mathbb{R}^n$, how can I ...
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In the book Topics in Differential geometry, Peter W. Michor defines the Fermi charts for a Riemannian manifold as follows. Let $(M,g)$ be a Riemannian manifold. For simplicity, I assume that $M$ is ...
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### Geodesics in geodesic balls

It is well-known that in a geodesic ball centered at $p$, the radial geodesic between $p$ and $q$ is the unique minimizing curve. I'm trying to follow the proof of this given in Cheeger & Ebin (...
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### Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
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### The set of Riemannian metrics on a submanifold of $\mathbb{R}^{n}$

Consider a subset $U \subset \mathbb{R}^{n}$. Clearly, $U$ can be considered as a smooth ($n$-dimensional) submanifold of $\mathbb{R}^{n}$. A Riemannian metric on $U$ is a smooth map $g$ which ...
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### Property of geodesic in surface of revolution in $R^3$ [on hold]

It is a question of my homework , I really don't know how to start it .
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### Connection between focal points and singularities of the normal exponential map

I am looking for nice references on focal points of Riemannian submanifolds. In particular, I would like to see a proof for the connection between focal points and singularities of the normal ...
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### A exercise of Riemannian geometry . [closed]

In picture below,I don't know how to start the second question . It is obvious that the isometry of $R^3$ keep the dimension , so there exist such isometry. But seemly, it is too simple . Besides, ...
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### How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric?

How to prove that two compact 1-dim Riemannian manifolds with same length must be isometric ? I know the compact 1-dim manifold must be homeomorphism to $S^1$ , but how to do a specific isometric ?
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### A Riemannian manifold with constant sectional curvature is Einstein. [closed]

A Riemannian manifold with constant sectional curvature is Einstein. Why? It's true the inverse?
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### About geodesics in product manifolds

My question is about the answer to this post : Geodesics on the product of manifolds If $(M_{1},g_{1})$, $(M_{2},g_{2})$ are Riemannian manifolds and if $\nabla^{1}$ (resp. $\nabla^{2}$) denotes the ...
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### Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
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### Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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### Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
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### Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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### Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow? I am familiar with the hackneyed expressions like "Ricci Flow is a non-...
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### Integration of differential form on ellipsoidal surface with singularity in origin

As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$\int_M\omega=\int _{\partial B_1(0)} \omega$$ I think there is a ...
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### How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
given the set $$M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\}$$ $m\in \mathbb{R}$, $\Omega$ is a bounded piecewise smooth domain in $\mathbb{R}^n$. also denote by $u(t)$ a map: $u(t):(0,T)\to M$ ...
I am reading some notes on black holes, and am confused by this definition of Kaluza-Klein asymptotic flatness: If a spacetime $(M, \mathbf{g})$ contains a spacelike hypersurface \$\mathscr{I}_{ext}...