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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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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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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Principal curvatures from parametrisation

Let $M^2$ be an immersed surface of the standard sphere $S^3$ with unit normal $\eta : M \to \mathbb{R}^4$ (tangent to $S^3$). Given a point $p \in M$ and a parametrisation $\varphi : U \subset \...
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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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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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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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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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19 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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Property of geodesic in surface of revolution in $R^3$

It is a question of my homework , I really don't know how to start it .
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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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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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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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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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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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A exercise of Riemannian geometry . [on hold]

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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1answer
17 views

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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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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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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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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sectional curvature, ricci tensor and scalar curvature of the hyperbolic space [closed]

Who can help me to compute the sectional curvature, Ricci tensor and the scalar curvature of the hyperbolic space $H^3$ ? Thanks!!
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1answer
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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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59 views

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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27 views

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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26 views

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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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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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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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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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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Help with derive geodesic equation

Let $(M,g)$ be a pseudo-riemannian manifold and $p,q\in M$. Suppose $\alpha:[a,b]\to M$ a smooth curve on $M$ such that $\alpha(a)=p$ and $\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{...
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Relearning differential geometry

I will shortly describe my situation and than formulate the problem. From around year I am working under supervision of my professor on master thesis in differential geometry (mainly discussion of ...
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A parallel transport around the Earth

Assume the Earth to be a 2-sphere. I start walking from the point $(\theta_0,\varphi_0)$ around the earth while my body heading north all along the way. In simple words, I walk east/west (It doesn't ...
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Is the norm of tensor fields just Hilbert-Schmidt norm/ generalized $L^p$-norm?

As I am rather comfortable with functional analysis language and was new to Riemannian geometry, I am curious when inspecting the norm of Ricci tensor, which is: $$|\mathrm{Ric}|^2=g^{ij}g^{kl}R_{ik}...
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How interpret the dual lattice $\Gamma^*$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \...
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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}...
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How is defined the inner product $g_p$ on $T_p \mathbb{R}^n/\Gamma$ at the point $p$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori. The lattice acts on $\mathbb R^n$ by $$γ(x)...
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Cauchy horizon of a future Cauchy hypersurface

I'm studing on the book Semi-Riemannian geometry by O'Neil. I'm tryng to understand the proof of the Hawking's singularity theorem (theorem 55A in the book). What I don't understand is why if $S$ ...
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1answer
25 views

Divergent Curves and Complete Manifolds

I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold $M$ is a curve $\alpha: [0, \infty) \to M$ ...
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When are embeddings into Euclidean space unique up to ambient isometry?

Suppose I have a Riemannian smooth manifold $M$ and a smooth isometric embedding $M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of $\mathbb{R}^n$? If not, ...
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Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group?

Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology. ...
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Is the variational field orthogonal to the velocity of the geodesic?

Let $\gamma(t)$ be a geodesic on a Riemannian manifold. Let $f(s,t)$ be a variation about $\gamma$. Is it always true that the variational field $\frac {\partial f}{ \partial s}(0,t)$ is always ...
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Existence of the lift of a curve

Let $(M, g)$ be a complete Riemannian manifold and let $(\tilde{M}, \tilde{g})$ its universal cover. Let $\pi : \tilde{M} \to M$ be the covering map. Let $\gamma : I \to (M, g)$ be a smooth curve ...
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Line in product mainifold

Let $(M_1, g_1)$ and $(M_2, g_2)$ be two complete Riemannian manifolds and consider the product $(M, g) = (M_1 \times M_2, g_1 + g_2)$. Let $\gamma : \mathbb{R} \to (M,g )$ be a line. I can write $t ...
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Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
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Understanding Energy minimization and poisson equation

Let $M$ be a Riemmanian manifold and $X$ be a vector field thereon. My question is why are these two problems equivalent?: \begin{equation} \operatorname{argmin}_{\phi}\int_M |\nabla \phi - X|^2 \...