(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

learn more… | top users | synonyms

0
votes
0answers
45 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
2
votes
1answer
45 views

Is a connected compact Riemannian manifold of dimension 1 unique?

The tiles says almost everything. It is known that a connected compact topological manifold of dimension 1 is isomorphic to $S^1$. What if we replace "topological" by "riemannian"?
0
votes
0answers
20 views

Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
3
votes
1answer
92 views

Connections in non-Riemannian geometry

In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of ...
2
votes
1answer
53 views

Extension of smooth maps at a cusp

There is a short remark in deCarmos "Riemannian Geometry" (p. 67) and I wonder about the condition that the vertex angles must be $\neq \pi$. If $s_1$ and $s_2$ are two differentiable maps on an ...
0
votes
1answer
29 views

Do we need a metric to define plurisubharmonic functions?

There are various notions of 'harmonicity' on various manifold. Sometimes, I am counfuesed by the definitions. For real manifold, the harmonic manifold is defined by $\Delta f=0$, where $\Delta$ is ...
1
vote
0answers
54 views

Special expression for 3-linear symmetric map $T(X,Y,Z) = \langle -Jh(X,Y),Z \rangle$ [Ejiri]

For a project in Riemannian Geometry, I have been working out the details of a paper by Ejiri (http://www.ams.org/journals/proc/1982-084-02/S0002-9939-1982-0637177-8/S0002-9939-1982-0637177-8.pdf) ...
1
vote
0answers
33 views

Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
1
vote
0answers
53 views

All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
1
vote
1answer
32 views

Hyperbolic (and related) structures on open unit disk

I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
1
vote
1answer
67 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
2
votes
1answer
89 views

When is the Hessian contracted with a vector field a closed form?

Suppose M is a Riemannian manifold and $g, f : M \rightarrow \mathbb{R}$ are smooth functions. When is the $1$-form Hess$^f(\nabla g, -)$ closed? I'm looking for simple conditions involving f,g and ...
3
votes
2answers
92 views

About two notions of holonomy

I have found something called "holonomy" in two apparently different contexts: Let $M$ be a smooth manifold, $E\to M$ a vector bundle and $\nabla $ a connection on $E$. Then you have a notion of ...
4
votes
2answers
60 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
0
votes
0answers
10 views

Existence of biregular cover on a Riemannian manifold with a codimension one foliation.

Suppose we have a (psaudo-)Riemannian manifold $(M,g)$ of dimension $m$ and that a codimension one foliation $\mathcal{F}$ exist for $M$. By Frobenius theorem this foliation is in one-to-one ...
1
vote
0answers
34 views

Volume of the ball in smooth Riemannian manifold

If V(x,r) is the volume of a ball B(x, r) on a smooth Riemannian manifold, for fixed point x, is the function V(x, r) continue or differential for radius r?
0
votes
1answer
45 views

Length Minimizing Properties of Geodesics on Surfaces?

Can anyone recommend me some nice references about lengh minimizing properties of geodesics? I'm looking for a treatment in the case of surfaces, but more general viewpoints will also be welcome. ...
0
votes
0answers
24 views

There are no conjugate points on a surface with negative Gaussian curvature?

I'm trying to understand the following theorem about conjugate points: Theorem. Let $M$ be a complete surface with Gaussian curvature $K\leq 0$, then there are no conjugate points on $M$. Proof: Let ...
1
vote
1answer
68 views

Can I construct an affine connection on a Riemannian manifold from arbitrary Christoffel Symbols?

The question is rather simple. All my definitions are as in Do Carmo's "Riemannian Geometry". If $M$ is a Riemannian Manifold, can I construct an affine connection $\nabla$ on it by setting, for all ...
2
votes
1answer
45 views

Computing the volume element of an oriented Riemannian manifold

I'm reading Gallot-Hulin-Lafontaine, and in section 2.7 they say they following: I wanted to check that the second $v_g,$ given in a local oriented chart, satisfied the first property. So I ...
2
votes
1answer
142 views

Why do these geometric assumptions imply these statements about relative homology?

I'm reading the paper Coverage in sensor networks via persistent homology. As in the paper, let $\mathcal{D}$ be a bounded domain in $\mathbf{R}^d$. We make the following assumptions: A5 The ...
1
vote
0answers
30 views

Isometries and geodesics in projective plane using covering

We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map ...
0
votes
1answer
43 views

Derivative of riemannian metric

I dont understand the following detail $$ \frac{1}{2} \int_a^b \frac{d}{dt}(g(X,X)ds = \int_a^bg(\nabla_YX, X)$$ Here $X = d\phi (\partial/\partial s)$ and $Y =d\phi (\partial/\partial t)$. Where ...
2
votes
2answers
72 views

Ricci Soliton geometric meaning

I wonder what is the geometrical, intuitive meaning of a Ricci soliton on a manifold. The definition that I use is as follows. $V$ is a vector field on the manifold, $g$ is a Riemannian metric. ...
2
votes
1answer
76 views

Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the ...
2
votes
1answer
66 views

Intuition/visualization for a non-flat connection

I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth ...
1
vote
1answer
36 views

For Green's function of $\Delta-c$, how to show $\int_{X}G(x,y)(\Delta-c)f(y) dy=\pm f(x)$?

Let $X$ be a compact Riemnnian manifold and $\Delta$ the Laplacian. Suppose that $G(x,y)$ be the Green's function of the elliptic operator $\Delta-c$ for a positive constant $c$. I think the ...
5
votes
0answers
119 views

Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each ...
2
votes
1answer
48 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
1
vote
1answer
30 views

Question on Normal Coordinates

I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to. ...
1
vote
1answer
62 views

Multi Index Dirac delta function

If we assume the following result: $$\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k, \rho}_{\beta_1,\beta_2,\cdots , \beta_k, \rho} = (n-k)\delta^{\alpha_1,\alpha_2,\cdots , ...
0
votes
1answer
62 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
5
votes
1answer
84 views

Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
0
votes
0answers
9 views

Short-time representation of variations of metrics on principal bundles via exp?

Let us consider a principal $G$-bundle $P\longrightarrow M$ together with an $H$-reduction $s$, where $H$ is a maximally compact Lie subgroup. As an $H$-reduction, $s\in\Gamma(M,P/H)$, hence we can ...
1
vote
1answer
56 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
1
vote
2answers
61 views

Isometries and inner product in Hyperbolic SPACE (H3)

Well, the title says what I need to know/understand, when I studied upper half plane I remember that isometries are Mobius transformations (If I am not wrong), now I have no clue about it. Thanks for ...
2
votes
0answers
57 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
1
vote
0answers
10 views

Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

Let $M$ be a compact Riemannian manifold. Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense?
0
votes
1answer
15 views

Density of bounded functions in $L^1(0,T;L^1(M))$?

Let $u \in L^1(0,T;L^1(M))$ where $M$ is a compact Riemannian manifold. Is it possible to find $u_n$ such that $u_n \to u$ in $L^1(0,T;L^1(M))$ and $u_n$ are bounded everywhere or almost everywhere on ...
7
votes
2answers
123 views

Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) ...
3
votes
2answers
100 views

Equivalent definitions of vector field

There are two definitions of a vector field on a smooth manifold $M$. A smooth map $V:M \rightarrow TM, \forall p \in M:V(p) \in T_p M$. A linear map $V:C^{\infty}(M) \rightarrow C^{\infty}(M), ...
0
votes
1answer
37 views

Problem in proving the property of Lie bracket of vector fields

Let $M$ be a Riemannian manifold, $f \in C^{\infty}(M)$, $X,Y$ vector fields on $M$. Then i have to prove $[X,f\cdot Y]=f\cdot [X,Y]+X(f)\cdot Y$. First i use the definition of Lie bracket: $[X,f\cdot ...
4
votes
1answer
70 views

The Levi-Civita connection in infinite dimensions

Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?
4
votes
1answer
49 views

“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
4
votes
1answer
59 views

Gauss curvature of C^2 surfaces

In do Carmo's book on Differential Geometry of Curves and Surfaces, the proof of theorema egregium, that the Gauss curvature of a surface immersed in $\mathbb{R}^3$ is invariant under local ...
0
votes
2answers
37 views

geodesic based on fixed points

Question: For any points $p,q\in M$, does there exist a geodesic curve connecting $p$ and $q$? Let $M$ be some constant curvature space, like $\mathbb R^n$, $\mathbb S^n$, $\mathbb H^n$. The answer ...
2
votes
1answer
18 views

Riemann integral and homoemorphism

I am wondering what happens if I have the following setup: I have a homeomorphism: $\phi$ from the unit sphere to the unit cube. I know that the characteristic function of the unit sphere is Riemann ...
2
votes
1answer
112 views

Totally geodesic and autoparallel

Let $M$ be a Riemannian manifold. A submanifold $N$ of $M$ is totally geodesic if every geodesic in $N$ is also a geodesic in $M$. On the other hand, $N$ is an autoparallel submanifold of $M$ if ...
0
votes
1answer
49 views

When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
0
votes
0answers
40 views

Strictly convex boundary of Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. What does it mean to say that the boundary is convex and strictly convex? I can find definitions of ...