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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37 views

How to distinguish between arc length and arc length parametrisation?

I am trying to understand and distinguish the difference between arc length and arc length parameterisation. The first thing, how do denote the $\text{arc length}$ and $\textit{arc length ...
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Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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90 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
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Reflection relating two subspaces

Let $S_1, S_2 \subseteq \mathbb{R}^n$ be two linear $k$-dimensional subspaces. Does there always exist a hyperplane $H$ such that $S_1 = R_H S_2$, where $R_H$ denotes the orthogonal reflection across $...
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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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88 views

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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66 views

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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17 views

Principal curvatures from parametrisation

Let $M^2$ be an immersed surface of the standard sphere $S^3$ with unit normal vector field $\eta : M \to \mathbb{R}^4$ (tangent to $S^3$). Given a point $p \in M$ and a parametrisation $\varphi : U \...
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sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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22 views

Is the set where the exponential map is defined an open subset of $TM$?

Let $M$ be a connected Riemannian manifold. Define $O=\{(p,v) \in TM|\, \,exp_p(v) \text{ is defined} \}$. Is $O$ an open subset of $TM$? I know that for every point in $M$, there is a neighbourhood $...
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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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1answer
30 views

About the Fermi charts

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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20 views

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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equidistant hypersurface

Let $M$ be a Riemannian manifold and $f: M \rightarrow \mathbb{R}$ with $ \parallel$ grad $f \parallel = 1 $. Let $ S = S_{t_{0}} = f^{-1} (t_{0})$ be a regular hypersurface. We define $\gamma: S \...
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Distance in submanifold vs ambient manifold

In Audin-Lanfontaine "Holomorphic curves in symplectic geometry", we need the following condition (Def. 4.7.1, p.182 by Sikorav): Let $L$ be a submanifold in a Riemannian manifold $(W,g)$. We want ...
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Do I have the right idea about affine connections?

On a smooth manifold $M$, a vector field is a smooth map $X : M \to TM$, where $TM$ is the tangent bundle of $M$. If $\chi(M)$ denotes the space of vector fields on $M$, an affine connection $\nabla$ ...
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210 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
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Metric and harmonic map (or map between manifolds)

There are many questions about what metric can be placed on a given manifold . For example , place a metric with non-negative curvature. As I know , the Gauss-Bonnet theory is useful in this question....
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1answer
419 views

How can i find this Basis in $\mathbb{R}^n$?

Define a Pseudo-Riemannian Metric $g$ in $\mathbb{R}^{n+1}$ by $g(u,v)=-u_0v_0+u_1v_1+...+u_nv_n$, where $u=(u_0,u_1,...u_n)$. Let $\eta\in\mathbb{R}^{n+1}$ be a vector such that $g(\eta,\eta)=-1$. Is ...
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free homotopy class of closed paths in a compact Riemannian manifold

Suppose that $M$ is a compact Riemannian manifold and that $\gamma$ is a closed path in $M$ which is assumed to be continuous but not necessarily piecewise smooth. Must the free homotopy class of $\...
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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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Torsion and curvature of a linear connection

Could you help me to solve the following problem ? Let $M$ a parallelizable manifold of dimension $n$, {$E_1$,...,$E_n$} a global frame of $M$. Let $X$,$Y$ a vector fields on $M$ with $Y= \sum_{i=1}^...
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Alternative characterization of complete metric space

Let $(X,d)$ be a metric space. It is complete if every Cauchy sequence for $d$ on $X$ is convergent. I've heard an alternative definition of completeness for $(X,d)$: it is complete iff the ...
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Green's Function for Laplacian on $S^1 \times S^2$

As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a ...
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When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
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1answer
30 views

What is $u^{-1}TN$ with $u: M\rightarrow N$ be a smooth map

As picture below, $u\in C^\infty(M,N)$, $(M,g)$ and $(N,h)$ are two smooth Riemannian manifold. I don't know what mean the $\frac{\partial }{\partial y^1} \circ u$ , it is $\frac{\partial u}{\partial ...
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1answer
48 views

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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63 views

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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1answer
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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 ...
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integral constraint induce a manifold on Sobolev space

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: $u(t):(0,T)\to M$ ...
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1answer
46 views

Intuition behind definition of K-K asymptotic flatness

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}...