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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Simple question about geodesic in Riemann manifold

I was read my book of Riemannian Geometry and the book says the follow: " A parameterized curve $\gamma:I\to M$ is a geodesic in $t_{0}\in I$ if $\dfrac{D}{dt}\left(\dfrac{d\gamma}{dt}\right)=0$ in ...
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0answers
128 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
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1answer
16 views

Orientable on almost complex manifold

I have troubles trying to prove almost complex two-dimensional manifold is orientable. Let I is complex structure on two-dimensional manifold M. Fix a basic $X_1,IX_1$ in each $T_xM$. Easy to see ...
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1answer
72 views

The cylinder does not embed into $\Bbb C^n$

The cylinder $\Bbb R\times S^1$ can be viewed as a complex manifold with a flat metric by viewing it has the quotient $\Bbb R\times\Bbb R/\Bbb Z$, where $\Bbb R\times\Bbb R=\Bbb C$. (In fact it makes ...
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39 views

Inclusions maps, parameterisation and charts

So it makes sense that an inclusion map $$\iota : S \longrightarrow S \subset M $$ maps $$ p \mapsto p $$ But how do you construct these guys in the context of manifolds? To my understanding, if $S$ ...
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1answer
36 views

Parallel transport along a closed geodesic

It do Carmo, in exercise 9.4, it is claimed that parallel transport along a closed geodesic in an even-dimensional orientable manifold "leaves a vector orthogonal to the geodesic invariant." So, let ...
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1answer
49 views

No conjugate points on $S^1\times \Bbb R$

Lee claims in his book that $S^1\times \Bbb R$ (considered as a submanifold of $\Bbb R^3$) admits no conjugate points along any geodesic. I am struggling to make that rigorous. Being conjugate along ...
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0answers
22 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: ...
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1answer
25 views

Informations about the cut-locus of a closed geodesic

Let be $(M^2,g)$ a closed riemannian manifold and $c:[0,L]\to M$ a simple closed geodesic on $M$. For each $s\in [0,L]$, let be $n(s)$ a unit normal vector field along to $c(s)$ and $\beta(s)$ the cut ...
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1answer
46 views

Eigenvalues of shape operator and of curvature on second exterior power

Terminology note In the following, a scalar product will be a symmetric bilinear form, and a euclidean scalar product will be a positive definite scalar product. This is the terminology used by my ...
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2answers
57 views

On the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$

Let $\exp$ be the exponential map on the Riemannian manifold M and $O$ is its domain in $TM$. Consider the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$, where $\pi$ is the ...
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7answers
349 views

On a definition of manifold

In the book Mathematical Masterpiece, on page 160, the authors wrote that A manifold, in Riemann's words, is a continuous transition of an instance I know a manifold is something glued by ...
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2answers
60 views

Three dimensional spherical excess formula

We all know the spherical excess formula: in a unit sphere, the area of a geodesic triangle is equal to the exceeding from $\pi$ of the sum of the three angles of the triangle. Is there a similar ...
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1answer
33 views

Isometric action on $S^n$

Let $S^n$ be the n dimensional sphere. For $n=2k+1$ odd, we identify $S^n$ as subset of $\mathbb{C}^{k+1}$. Furthermore we can define the action $$\Psi: S^1 \times S^n \to S^n, (c,(z_0, \dots, z_n)) ...
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1answer
37 views

Scalar curvature of metric? [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = f(x)\,dt^2 + dx^2.$$What is the scalar curvature, $R$, of this metric?
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1answer
29 views

If $M_{\alpha\beta}\ge0$ , how to show $M^\#\ge0$?

We say $M_{\alpha\beta}\ge0$ if $M_{\alpha\beta}v^\alpha v^\beta\ge0$ (sum over $\alpha,\beta$) for all vectors $v=\{v^\alpha\}$. If $M_{\alpha\beta}\ge0$ ,how to show $$ M^\#\ge0 ~? $$ Relative ...
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1answer
25 views

Identity surrounding Killing vector field on a spacetime $\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d$

Let $w^a$ be a Killing vector field on a spacetime $(M, g_{ab})$, i.e., $w^a$ satisfies $\nabla_{(a}w_{b)} = 0$. I hypothesize that$$\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d,$$but I am not sure how I ...
4
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1answer
40 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...
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1answer
19 views

Find a isometric immersion of the Torus $T^{n}$ on $\mathbb{R}^{2n}$.

Find a isometric immersion of the "plane" Torus $T^{n}$ on $\mathbb{R}^{2n}$. Isometric Immersion, let $f:M^{n}\to N^{n+k}$ a immersion, i.e., $f$ is differentiable and $df_{p}:T_{p}M\to ...
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1answer
27 views

About totally umbilical hypersurfaces

Suppose $\tilde{M} \subset M$ is a hypersurface sitting inside a Riemannian manifold $(M,g)$. The second fundamental form of $M$ evaluated on $u,v \in T_pM$ is denoted $II(u,v)$ and defined as the ...
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1answer
35 views

Covariant Contravariant approach for Tensors

I'm reading a book on Geometry from the '70s and when speaking about Tensors it defines them starting from the covariant and contravariant commutation rule. I know this definition was quite widespread ...
3
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1answer
58 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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1answer
32 views

Minimizing curves are geodesics

Let $(M,g)$ be a Riemannian manifold. I want to prove the following claim: Let $c:[0,1]\to M$ be a smooth curve from $p$ to $q$ such that $L(c)=d(p,q)$. Then $c$ is, up to reparametrization, a ...
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0answers
23 views

Isometries between the hyperboloid and the plane?

What is the map from $\mathbb{H}^2$ to $\mathbb{R}^2$ that preserves the pairwise geodesic distances in one as closely as possible to the pairwise geodesic distances of the images in the other? ...
10
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1answer
106 views

Identity in general relativity, not sure if true or not

Let $(M, g_{ab})$ be a spacetime and define a new metric, $\tilde{g}_{ab}$, on $M$ by $\tilde{g}_{ab} = \Omega^2 g_{ab}$, where $\Omega$ is a smooth, positive function. Let $\nabla_a$ denote the ...
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1answer
35 views

Using Koszul's formula to compute $\nabla_X Y$, where $\nabla$ is the Levi-Civita connection

Main question: I am following the answer in this question: computing Riemannian connection and Killing fields (very basic). He calculates $$ g(\nabla_X(Y),X) = 0 \\ g(\nabla_X(Y),Y) = 0 \\ ...
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1answer
36 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
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1answer
24 views

Does it necessarily follow that the integral curves of $k^a$ are null geodesics?

Let $f$ be a function on a spacetime $(M, g_{ab})$ whose gradient, $k_a = \nabla_a f$, ie everywhere null, i.e., $k_ak^a = 0$ throughout $M$. Does it necessarily follow that the integral curves of ...
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0answers
55 views

Showing that a specifik structure on $\mathbb{R}^2$ is a complete metric

I have smooth maps $f,h:\mathbb{R}\rightarrow (0,\infty )$ with $f(t)\geq k$ and $h(t)\geq \frac{1}{\mid t\mid}$ for all $\mid t\mid >c$ for some$k,c>0$. I want to prove that ...
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1answer
33 views

When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold?

Asked here too: http://mathoverflow.net/questions/235178/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz A reference on totally geodesic submanifold (TGS): ...
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0answers
47 views

Injectivity Radius of Surface Level

Can tip this problem. I did a part , but could not complete. They gave me a tip to complete , but could not. I thank the help . $\mathbf{Problem}$ Let $\, \, \, f: \mathbb{S}^{n+1} \rightarrow ...
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0answers
41 views

Riemannian metric on a level set of a smooth function on a manifold

Also asked here: http://mathoverflow.net/questions/235163/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold Let $(M,g)$ be a finite or infinite dimensional Riemannian manifold. Let ...
2
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1answer
31 views

Computing Sectional Curvature on Hyperbolic Plane

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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1answer
19 views

Condition for a one-parameter family of maps to be isometries

Given two smooth manifolds with Riemannian metric $(X,g)$ and $(Y,h)$ and a smooth map $f: X \to Y$ I understand that we define $\phi$ to be an isometry if $f^* g = h$. I thought I understood this ...
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2answers
59 views

Algebraic topology & Riemannian geometry project idea?

I'm taking a first course on Riemannian geometry this semester. For a final project, I would like to do something that involves algebraic topology. However, the only results I know in algebraic ...
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0answers
30 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
3
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1answer
37 views

Laplacian of a distance function on a Riemann manifold

For some reasons I need to show the following fact. Let $(M, g)$ be a Riemannian manifold. Let $U \subset M$ be an open set and $r: M \to \mathbb{R}$ a smooth distance function. Let us assume ...
2
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0answers
49 views

Computing connections on manifolds

Let $\nabla$ denote the Levi-Civita connection on the following manifold in $\mathbb{R}^3$ with Riemannian metric $g$ as follows: \begin{equation} \mathcal{H}^3=\lbrace (x,y,z)\in \mathbb{R}^3 \mid ...
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0answers
45 views

Motivation for Non-Euclidean geometry: relativity

I'm looking for references to motivate the study of non-Euclidean geometry. In particular I would like something about relativity. I do not want texts to learn non-Euclidean geometry, only ...
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1answer
28 views

Extending parallel 1 forms to harmonic forms on a compact set

Based on this question from Peterson's Riemannian Geometry: Let $(M,g)$ be an n-dimensional connected Riemannian manifold that is isometric to Euclidean space outside some compact subset $K ...
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1answer
34 views

Proving that Levi-Civita connection is preserved by isometries

I am trying to prove that given two Riemannian submanifolds $S,S'$ with Levi-Civita connections $\nabla , \nabla'$ and an isometry $f$, then $$ Df(\nabla_XY)=\nabla'_{X'}Y' $$ where, ...
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1answer
60 views

Orthonormality of vector fields

I want to show that the following Riemannian manifold with given metric $g$ the following vector fields are orthonormal at every point $p$ of the manifold. Let $\mathcal{H}^3=\lbrace (x,y,z) \in ...
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1answer
33 views

Computing the Lie brackets of vector fields on a specific Riemannian manifold

Let $\mathcal{H}^3=\lbrace (x,y,z) \in \mathbb{R}^3 \mid z>0\rbrace$ be equipped with the Riemannian metric: \begin{equation*} g=\frac{dx^2+dy^2+dz^2}{z^2} \end{equation*} And consider the vector ...
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2answers
42 views

Levi-Civita Connection and vanishing christoffel symbols

Is there a good way to guess for what indices christoffel symbols, $\Gamma_{ij}^k$ vanish in general? For example, when calculating the Levi-Civita with spherical coordinates for a sphere most ...
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0answers
35 views

Dual basis cotangent space

I have been given the unitary sphere in the Euclidean space. $$F(\theta, \phi) =(\sin\theta \cos\phi, \sin\theta \sin\phi,\cos\theta)$$ I'm asked to show that the dual base of $E_1=F_*(\partial ...
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1answer
48 views

Reference about Karen Uhlenbeck

When I read Hamilton's 'FOUR-MANIFOLDS WITH POSITIVE CURVATURE OPERATOR',I am curious the details about Karen Uhlenbeck trick. But I can't find suitable reference to read it . What should I read ...
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1answer
34 views

Gaussian curvature versus sectional curvature

I was studying https://en.wikipedia.org/wiki/Gaussian_curvature (exact version https://en.wikipedia.org/w/index.php?title=Gaussian_curvature&oldid=709607678 ) and there it says: (bold added) ...
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1answer
58 views

A notion of nonpositive curvature for general metric spaces

The proof of the following result should be done by using the second variation formula of geodesics but I do not know how to start or what is the main idea of the proof. (Lemma 3.7 in the paper: A ...
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1answer
26 views

Counting independent components of Riemann curvature tensor

I'm having some trouble understanding the counting procedure for the number of independent components of Riemann curvature tensor $R_{iklm}$ in 4D spacetime. (The answer is supposed to be 20, but I'm ...
4
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1answer
39 views

Coordinates on a Riemannian manifold given by a distance function

I am currently studying the book "Riemannian Geometry" by Petersen. Defintion: Let $(M, g)$ be Riemannian manifold and let $U \subset M$ be an open set. A function $r : U \to \mathbb{R}$ is said ...