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 ...

learn more… | top users | synonyms

0
votes
0answers
27 views

Dimensionality of tangent vectors in R^2

I am puzzled with the following problem: given a tangent vector (a d/dx) in the Euclidean plane R^2 with "a" a dimensionless scalar, the dimensionality of this vector is, I suppose, 1/[lenght] and ...
1
vote
1answer
50 views

Does $ \int_{M} || \nabla_{M} f||^2 dS = \int_{\Omega} ||\nabla_{\mathbb{R^n}} (f \circ \Phi)||^2g d\lambda,$ hold?

Let $\Phi: \Omega \subset \mathbb{R}^n \rightarrow M$ and $M$ a euclidean manifold. Is it then correct that $$ \int_{M} || \nabla_{M} f||^2 dS = \int_{\Omega} ||\nabla_{\mathbb{R^n}} (f \circ ...
3
votes
2answers
90 views

Zero Sectional Curvature implies exp is a local isometry

Im studying DoCarmo's book Riemannian Geometry, the first problem of the chapter 5 (Jacobi Fields) states that If $(M,g)$ is a riemannian manifold with sectional curvature identically zero, show that ...
2
votes
0answers
47 views

Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
1
vote
0answers
23 views

Dimension of scalar solutions to these self-dual/anti-self-dual equations

Let $M$ be a 4-dimensional Riemannian manifold. Let $\kappa$ be a 1-form. I look for solution function $\phi$, such that there exists functions $\alpha$ and $\beta$ \begin{equation} {\left( ...
3
votes
1answer
62 views

Showing two forms on a manifold are equal

Let $\alpha$ and $\beta$ be two forms on a manifold $M$. To show that they are equal, does it suffice to show that for arbitrary $p\in M$ there exists some chart such that $\alpha_p=\beta_p$. I was ...
0
votes
1answer
40 views

How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
4
votes
1answer
42 views

Geodesic flow on a non-complete Riemannian manifold with constant positive curvature

I am trying to understand the geodesic flow on the following 2-dimensional Riemannian manifold $M$. As a set, $M$ is the interior of the standard 2-simplex, $$M=\{(x,y)\in\mathbb{R}^2\mid ...
1
vote
0answers
31 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
0
votes
1answer
116 views

Riemannian metric induced by metric

This seems a very basic and useful construction, and yet I cannot find any reference for it. So my questions are, 1) Is the following definition correct? 2) Is there a simpler construction? 3) Do you ...
1
vote
1answer
42 views

Two isometries that have same value and differential at some point are the same.

I also have trouble in this problem: Let $f, g$ be two isometries of a connected Riemannian manifold $(M, g)$. If $f(p)=g(p)$, $df_p = dg_p$, show that $f=g$. Any comment is expected. I know it ...
0
votes
1answer
27 views

Minimal geodesic on the real projective $n$-space

I have encountered this problem: Show that a geodesic of $(\mathbb RP_n, g_0)$ with $g_0$ being the metric given by the canonical metric on $\mathbb S^n$ via the $2:1$ Riemainnian covering, is ...
0
votes
1answer
44 views

Frame acting on a curve/Geodesic eqution

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
0
votes
1answer
15 views

Length shortening Riemannian metrics

I am looking for examples of Riemannian metrics such that the curve length under these metrics are always smaller than the length as measured in Euclidean space. It is just a question that popped into ...
1
vote
1answer
95 views

Exterior derivative of forms derived from a metric

Let $(M,g)$ be a Riemannian manifold. From $g$ and a fixed vector field $V$ we can derive the following two differential forms: A $1$- form $\alpha(X) = g(V,X)$, i.e. $\alpha = \iota_Vg$. A $2$-form ...
1
vote
1answer
44 views

Surfaces of constant curvature using the conformal method

I'm doing a study of surfaces with constant curvature which leads to solving the equation: $$\Delta\phi = -e^{2\phi}K_0$$ for a 2-dimensional metric with constant curvature such that rotation around ...
1
vote
0answers
42 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
1
vote
0answers
35 views

Is the Riemann curvature tensor the only tensor that can be constructed from the metric tensor and its first and second derivatives?

I am reading Gravitation and Cosmology by Steven Weinberg. On page 133, he says $R^{\lambda}_{\phantom{x}\mu\nu\kappa}$ is the only tensor that can be constructed from the metric tensor and its ...
0
votes
0answers
15 views

Stable geodesics

Consider a function defined on some space of smooth curves in a manifold (think of the "action functional"). I understand what a "critical point" of such a function is, but what is understood by a ...
0
votes
0answers
29 views

The Jacobian of the exponential along a geodesic

I am reading a paper that uses but does not define the following concept: what is understood by "the Jacobian of the exponential map along a geodesic (beetween two points)"? Is this only defined for ...
0
votes
1answer
38 views

Dot product of two cross products in $\Bbb R^3$ with general metric

I would like to find the generalized formula of the identity $$(A\times B).(C\times D)=(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C)$$ which holds in an Euclidian metric, within a general metric $g$ on ...
1
vote
0answers
31 views

Linearisation in direction of formal adjoint

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. The Ricci curvature can be viewed as a differential operator ...
3
votes
1answer
42 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
1
vote
0answers
46 views

Does covariant derivative commute with “generalized contraction”_ About the proof of 2nd Bianchi identity

I am reading the proof of second Bianchi identity on wiki. In the proof, it says the following condition must satisfy: $$((D_X R) (Y,Z)) (W) + R (D_XY,Z) W + R(Y,D_XZ) W + R(Y,Z) D_X W = D_X ...
1
vote
0answers
35 views

Prove the local expression of Riemannian curvature tensor

I try to prove the following expression of Riemannian curvature tensor: For local coordinate $\{x^i\}$, let $g_{ij}=g(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})$ and ...
1
vote
0answers
61 views

Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
0
votes
1answer
57 views

Riemannian distance induced by an elliptic differential operator?

consider a Riemannian manifold $(M,g)$ and consider a second order elliptic differential operator. I've read that each such operator induces a riemannian distance function. Unfortunately I couldn't ...
0
votes
2answers
60 views

If we don't need a Riemannian metric to compare length of vectors, why do we use metrics to measure curvature?

I read that, in the absence of a Riemannian metric tensor field, we can still measure how much a vector changes when parallel transported around a curve by comparing the initial and final vectors. ...
0
votes
0answers
36 views

harmonic function on manifold

Let M be a 2 dimensional manifold. $h:M\rightarrow R$ be a harmonic function from manifold to real line. G is group that act by isometry. $g*h(x)=h(g(x))$. Let $W=\{x|h(x)=t\}$ that is the level set ...
0
votes
0answers
20 views

Recommendable books to study the Selberg zeta function.

I've study on the Riemann zeta function and some zeta functions which have analytic properties directly. And now I want to know about the Selberg's zeta function which has some geometric properties. ...
1
vote
1answer
94 views

How are “scalar curvature” and “sectional curvature” related?

I was browsing wikipedia and was puzzeling about what is the difference between: "scalar curvature" https://en.wikipedia.org/wiki/Scalar_curvature and "sectional curvature" ...
1
vote
0answers
43 views

Schwarzschild half-plane and its geodesics

For some fixed $r_0>0,$ put the semi-Riemannian metric $$ ds^2=\frac{r_0-r}{r}dt^2+\frac{r}{r-r_0}dr^2 $$ on $\{(t,r)\in\mathbb{R}^2:r>r_0\}.$ I would like to show that the $r$-lines are always ...
1
vote
2answers
21 views

What exactly are the independent components?

What are the 20 independent, non zero components of the 4D Riemann curvature tensor? (Not how many, I know there are twenty, but specifically which components are non-zero?)
2
votes
1answer
65 views

Verification of product rule for covariant derivatives. Stuck on one step involving simplifying terms to yield zero.

I am trying to learn more about covariant differentiation. I'm specifically interested in physics applications, but I found this nice exercise in Misner, Thorne, and Wheeler's book Gravitation that I ...
0
votes
1answer
41 views

Projection of fiber bundle is a submersion

I'm just wondering about my proof for the following fact. I get the feeling it is almost trivial but I am still getting a feel for geometry and so it doesn't seem 'obvious' to me just yet. The ...
0
votes
1answer
69 views

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions:

Let C be an oriented plane curve with curvature k>0. Assume that C has at least one point p of self intersection, prove the following questions: a. There is another point $p_{0}$ such that the ...
0
votes
0answers
59 views

definition of derivative of a vector field

i try to define by myself the notion of differentiation of a vector field on a general manifold. I know that it is a classical subject and that there exist some answers as Lie derivative of a vector ...
2
votes
1answer
42 views

Riemann manifold with unbounded Laplacian

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its ...
8
votes
2answers
221 views

Equivalence of two distance function on a Riemannian manifold

Let $(M,g)$ be a closed connected $m$ dimensional smooth Riemannian manifold and assume that it is isometrically embedded in a Euclidean space $\mathbb{R}^q$ by $\iota:M\to\mathbb{R}^q$. $|\ast|$ ...
0
votes
0answers
57 views

Prove properties of induced connection from a Levi-Civita connection

I just learnt let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, ...
2
votes
1answer
38 views

What does $\langle R(x,z_i)x,z_i \rangle$ represent geometrically?

In his book Riemannian Geometry, Manfredo Do Carmo states the following on page 97: Let $x = z_n$ be a unit vector in $T_pM$; we take an orthonormal basis $\lbrace z_1,z_2,...,z_{n-1}\rbrace$ of ...
1
vote
1answer
51 views

Computation of the extrinsic curvature tensor for a warp drive metric.

In Miguel Alcubierre's renowned paper discussing a "warp drive" metric, he discusses the extrinsic curvature. Here is an extract. My questions are quite trivial to someone who understands the ...
2
votes
0answers
29 views

Induced connection is well-defined

Let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map. If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, then $$\forall ...
2
votes
1answer
28 views

$D_vX$ is completely determined by $X$ restricted on any curve $r$ with $r'(0)=v,r(0)=p$.

I want to show given $v\in T_pM$, then $D_vX$ is completely determined by $X$ restricted to any curve $r$ with $r'(0)=v,r(0)=p$. I have shown that if $r_1'(0)=r'_2(0)=v,$ then ...
1
vote
0answers
18 views

Recognizing regular distributions

By "regular" distributions I understand those Schwartz distributions that arise from locally-integrable functions. Are there ways of telling them apart from the non-regular ones? Does the set of those ...
3
votes
2answers
105 views

Equivalence for Christoffel symbol and Koszul formula

I am trying to show to define a Levi-civita connection, it's equivalent to define Christoffel symbols or define Koszul formula. $$ 2g(\nabla_XY, Z) = \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - ...
1
vote
1answer
35 views

An explicit Lorentzian metric on the Klein bottle

I want to construct an explicit Lorentzian metric on the (abstract) Klein bottle but have no idea where to start. Could someone please give me a hint and/or guide me in the right direction? Thanks.
1
vote
1answer
72 views

Dehn twist as isometries on hyperbolic surface

[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus] Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma $ I can realized a ...
0
votes
2answers
58 views

hyperbolic geometry and affine transformations

A very famous geometer recently commented to me that "hyperbolic geometries are the only geometries invariant under affine transformations". It is unclear to me what this comment even means. Can ...
4
votes
2answers
97 views

Can someone explain the basic idea behind the sectional curvature formula?

I found the following equation on Wikipedia here: \begin{equation} K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2} \end{equation} No explanation I ...