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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Continuous function with non-negative second derivative in the weak sense is convex

I am currently working through a section of Peter Petersen's Riemannian Geometry in which he talks about weak second derivatives of functions. I am trying to work through the details of why a function ...
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51 views

Help understanding John Lee's definition of curvature

On page 3 of his book Riemannian Manifolds, John Lee states the following If you want to continue your study of plane geometry beyond figures constructed from lines and circles, sooner or later you ...
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86 views

Characterisation of local affine diffeomorphisms

I've got a question about local affine diffeomorphisms between affine manifolds. There ist a good characterisation about affine diffeomorphisms of connected affine Mannifolds: Let $f,g\colon ...
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1answer
54 views

Non-constant rank of a smooth map and orthnormal basis in the normal bundle

Assume $M,N$ are two Riemannian manifolds and $f: M\rightarrow N$ is a smooth map. Suppose $dim M =m < dim N =n$. Let $\Sigma$ be the graph of $f$, that is, $\Sigma =(x, f(x))$ for $x\in M$. My ...
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Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
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52 views

Finding a frame for a vector bundle in a smooth manifold with a connection

I am trying to solve the following exercise: Let $P$ be a vector bundle over a smooth manifold $M$ with a connection $\nabla$, and let $p \in M$ . Show that there is an open set $U$ of $M$ with $p ...
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1answer
90 views

Is the tangent-cotangent isomorphism orientation preserving?

Consider $(M,g)$ a Riemannian manifold. Let's define $\varphi : TM\rightarrow T^{\ast}M$ by $\varphi(p,v):=(p,g(v,.))$, for $p\in M$ and $v\in T_{p}M$. Here, $TM$ stands for the tangent bundle and ...
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44 views

What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero ...
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76 views

Does a Riemannian metric allow definition of a tangent vector's length?

In Euclidean spaces, we define the Euclidean norm of a vector $\vec{x} = (x_1,x_2,...x_n)$ as $\|\vec{x}\|:=\sqrt{x_1^2+x_2^2+ \cdots +x_n^2 }$ Does the metric tensor field of a Riemannian manifold ...
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1answer
107 views

How can I measure the length of a curve using a Riemannian metric?

On page 35 in his book Riemannian Geometry, Manfredo do Carmo states the following: Giving a surface $S \subset \Bbb{R}^{3}$, we have a natural way of measuring the lengths of vectors tangent to ...
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1answer
80 views

Is $M \times (0,\infty)$ a manifold of bounded geometry?

If $M$ is a compact Riemannian manifold, is $M \times (0,\infty)$ a manifold of bounded geometry? I think it is, since $M$ is compact and $(0,\infty)$ is simply flat.
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100 views

Completeness implies geodesic completeness, a more conceptual way?

We know from Riemannian geometry that for Riemannian manifolds, completeness and geodesic completeness are equivalent, which is usually a consequence of Hopf-Rinow theorem. However, I'm considering a ...
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1answer
33 views

Definiton of a riemanian metric (smoothness)

In Ana Cannas de silva's book Lectures on Symplectic geometry he defines a positive inner product to be smooth when for any vector field $v$ the function $x \mapsto g_x(v_x, v_x)$ is smooth. Some ...
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66 views

Every possible choice of Christoffel symbols generate a valid connection

Does every possible choice of Christoffel symbols generate a valid connection? Or is there some restriction on them?
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75 views

In the standard definition of an affine connection, what does $\nabla$ stand for?

I am very confused about what $\nabla$ signifies when used to describe affine connections. In his book Riemannian Geometry, Manfredo do Carmo defines an affine connection as follows. Let ...
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77 views

Breaking down what the definition of an affine connection says

In his book Riemannian Geometry, Manfredo do Carmo defines an affine connection as follows: Let $\mathcal{X}(M)$ denote the set of all vector fields of class $C^{\infty}$ on $M$. Let ...
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1answer
247 views

Laplace-de Rham operator

Consider an operator $\partial = (-1)^k \star^{-1}\,d\star: \Omega^k(\mathbb{R}^n) \to \Omega^{k-1}(\mathbb{R}^n)$. Note that we equivalently can write $\partial = (-1)^{nk + n + 1} \star\,d\star$. ...
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1answer
164 views

Sufficient condition for $M$ to have constant curvature

I decided to keep my original question. However, I'm having trouble only in a part of it (check NOTE) Let's consider a Riemannian manifold $(M,g)$, with the Levi-Civita connection $\nabla$. I would ...
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1answer
63 views

Boundary points of a manifold

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ...
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70 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
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0answers
41 views

When does a pseudo-Riemannian manifold have an always positive norm Killing field?

When does a pseudo-Riemannian manifold have an always positive norm Killing field? (you may assume that the isometry group is of the form $SO(1,n)$ if necessary) In the context of general ...
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188 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
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85 views

At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
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2answers
114 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
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1answer
25 views

A question about sum of angles in a non-positive curvature Riemannian manifold

Suppose on a non-positive curvature Riemannian manifold,we have a geodesic triangle $\triangle abc$ ,and counterpart edges donates $\alpha,\beta,\gamma$. If now I get $$ a^2 \geqq b^2+c^2-2bc ...
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1answer
52 views

Ricci curvature version of Cartan-Hadamard theorem?

Is the following assertion true : If $M$ is a simply-connected manifold with $\operatorname{Ric}<0$ (or $\operatorname{Ric}\leq -k$ for $k$ positive) then $M$ is diffeomorphic to $\mathbb{R}^n$? ...
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78 views

Is Relativity a specific instance of Riemannian geometry?

If I am a mathematician and do not anything about Special/General Relativity, then should I study Riemannian geometry to learn Relativity? Is Relativity just an instance/example of some particular ...
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1answer
48 views

Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature

I am looking for examples of compact manifold of dimension $>2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help?
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1answer
71 views

Symmetry of Killing Vectors in Covariant Derivative

Several times, I've seen statements along the lines of "$\nabla_X Y=\nabla_Y X$ because $X$ is a Killing vector field." One example I found on Stack Exchange is here. I have yet to understand why ...
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1answer
90 views

Smoothly homotoping a sphere in $\mathbb{R}^3$

Start with the standard sphere $S^2$ and consider another (diffeomorphic) sphere $S$ such that there is a family of deformations of $S^2$ in $\mathbb{R}^3$ that ends in $S$. If $S$ is positively ...
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2answers
130 views

Conformal transformation of the divergence

Let $f$ be a smooth function on a $n$-dimensional Riemannian mainfold $(M, g)$, so that $\tilde{g} = e^{2f} g$ is a conformal transformation of $g$. I'm trying to show that the divergence operator ...
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1answer
321 views

Riemannian curvature tensor of product manifolds

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds. Let $% R_{1}$ and $R_{2}$ be the (1,3)-type Riemannian curvature tensors of $M_{1}$ and $M_{2}$, respectively. Finally, let $R$ be ...
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92 views

Cohomology in Differential Geometry

Below is a communicative diagram: $$\begin{array}[c]{ccc} ...
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53 views

What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
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1answer
74 views

Jacobi field and the metric

I'm reading about Jacobi fields lately, and have noticed some features of it (and it's derivative) with respect to the metric. Thinking about that, I had an non-based, purely intuitive thought that ...
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94 views

Closed geodesic loop on compact manifold

Let $M$ be a compact manifold (hence complete). Let $p$ be any point on $M$. Is it true that we can always find a geodesic loop based at $p$? If $M$ is non-simply connected it is true as each ...
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58 views

Surfaces (with boundary) in $\mathbb{R}^3$ conformal to the cylinder

Consider the usual cylinder $S^1 \times [0, 1]$ embedded in $\mathbb{R}^3$. I am interested in knowing what are the surfaces in $\mathbb{R}^3$ that are conformal to this cylinder. If this were a ...
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1answer
98 views

How to show that geodesics exist for all of time in a compact manifold?

Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form ...
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221 views

Covariant derivative of vector field along itself: $\nabla_X X$

Consider a vector field $X$ on a smooth pseudo-Riemannian manifold $M$. Let $\nabla$ denote the Levi-Civita connection of $M$. Under which conditions can something interesting be said about the ...
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1answer
34 views

Show that a parallel field has constant length.

Show that a parallel field has constant length (Riemannian-geometry). It is true for all connections?
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1answer
38 views

Find an example of n-dimensional differentiable manifold

Find an example of $n$-dimensional differentiable manifold whose points are not points of the variety $\mathbb{R}^{n}$
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30 views

Metric in normal coordinate

Using Gauss’s lemma we can write the metric in normal co-ordinate as $g(r, θ) = dr^2 + r^2h_{ij}(r, θ)dθ^i ⊗ dθ^j$ (where metric on $S^{n-1}$ is $\tilde {g}=dθ^i ⊗ dθ^i$). Now as $r \rightarrow 0$, ...
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1answer
131 views

Showing a property of a curvature tensor in $S^2$

Consider $S^2 \subset \mathbb{R}^3$. I need to show that if $$R_{ijkl} = -g(R(\partial_i,\partial_j)\partial_k,\partial_l)$$ is a curvature tensor in $S^2$ and $g$ is a metric also in $S^2$, then ...
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0answers
64 views

Variation of geodesic and Jacobi field

I was reading on Jacobi field of a geodesic, and noticed that given a geodesic $\gamma$ it is defined using the term of variation or family of geodesics $\gamma_s$ but never mentioned how to create ...
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1answer
72 views

Are there “interesting” examples of complete finite-volume non-compact Riemannian manifolds that are not non-positively curved?

Moreover, it would be good that, if $M$ is such an example, its universal cover $\tilde M$ is "highly" symmetric, i.e. the group G of isometries of $\tilde M$ is "big" (for example, transitive, or a ...
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1answer
61 views

Geodesic loops in Riemann homogeneous spaces

Let M be a Riemannian homogeneous space, i.e. the isometry group acts transitively. Prove: any geodesic loop (with possible angle at the starting point) is a closed geodesic (smooth at the starting ...
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1answer
410 views

Sectional curvature of product metric?

If $M$ and $N$ are Riemannian manifolds, can we relate the sectional curvature of the product Riemannian manifold $M \times N$ to those of $M$ and $N$? If both $M$ and $N$ have non-negative (or ...
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2answers
86 views

Riemann manifolds in relation to other classes of differentiable manifolds

I am trying to get an overview over the different categories of manifolds. In particular i have the following chain of inclusions: Riemann surfaces $\subset$ complex manifolds $\subset$ orientable ...
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1answer
223 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
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1answer
18 views

Out of plane cross section evolution of surfaces based on local geometry information

With this question I would like to kindly ask for feedback or general pointers to even remotely related works in regards to a challenge I face. Given a smooth surface $S$ $:\mathbb{R}^2\rightarrow ...