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

2
votes
1answer
179 views

Hessian matrix on Riemannian manifolds

$(M, g)$ is a Riemannian manifold, and $u$ is a smooth function on $M$, one says that under a normal coordinate system, $\operatorname{Hess}(u)_{ij} = (u_{lk} - u_h \Gamma^h _{lk})B^{li}B^{kj}$, where ...
2
votes
1answer
99 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
0
votes
1answer
80 views

quotient of 2-torus by antiholomorphic involution is annulus?

I would like to study what the quotient $$T^2 / \Omega $$ of a closed compact Riemann surface with $g=1$ handles, once a complex structure is chosen, over an antiholomorphic involution $\Omega,$ can ...
3
votes
1answer
118 views

Linear connection on a manifold: Math vs. Physics

I have learned some Riemannian Geometry in a strongly mathematical framework, precisely from the book "J.M.Lee - Riemannian Manifolds: An introduction to Curvature". Now I'm trying to learn ...
2
votes
2answers
106 views

Show that $a \wedge * b = g(a,b) \operatorname{vol}$

$\newcommand{\vol}{\operatorname{vol}}$ Let $\omega$ be a $p$-form on a Riemannian manifold $M^n$ with metric $g$ and let $\vol_{i_1,\ldots,i_n}=\sqrt{\lvert g\rvert} \epsilon_{i_1,\ldots,i_n}$ be a ...
1
vote
0answers
70 views

Suggestion for a good book that explain Cartan's Moving Frame and Riemannian Geometry

I'm studying Riemannian Geometry, and I'm having a lot of trouble with the book Riemannian Geometry and Differential Dorms both from do Carmo.And I would like a book with examples, calculations, if ...
1
vote
0answers
63 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
6
votes
1answer
190 views

Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
0
votes
1answer
66 views

How to find a dual frame at the sphere

Let $U=\{ (\theta,\phi,r):\theta \in \mathbb{R}, \phi \in ]0,\pi[,r\gt 0\}$, how I can find the moving frame. I thougt: Consider the parametization for $U$ $$(\theta,\phi,r)\mapsto ...
1
vote
1answer
142 views

Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
2
votes
0answers
39 views

Laplacian on a warped product.

Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the ...
2
votes
3answers
118 views

Can there be non trivial self-dual 5-forms on a 10-dimensional compact orientable manifold without boundary?

I am puzzled about the following. Let $(M,g)$ be a compact, orientable Riemannian manifold without boundary. We define the usual inner product $(,)$ for two r-forms $\alpha,\beta\in\Omega^r(M)$ by ...
3
votes
1answer
116 views

Harmonic function on Riemannian manifold

$M$ is a connected Riemannian manifold with $\Delta$ its Laplacian and $f$ is smooth function on $M$ such that $\Delta f=0$ and $f$ vanishes on some open set $U$ of $M$, then is $f$ identically $0$ on ...
2
votes
1answer
45 views

The choice of the eigenfunction of Laplacian

$M$ is a closed Riemannian manifold and $\lambda_1>0$ is the first nontrivial eigenvalue of $\Delta$. Can we find a eigenfunction $f$ of $\lambda_1$ such that $\mathop {\sup }\limits_M f - \mathop ...
2
votes
0answers
40 views

Are part of great circles geodesics on spherically symmetric manifolds?

While doing some study on geodesics on Riemannian manifolds, I learned that any geodesics on $n$-spheres are part of great circles. I then started wondering if that is true for any spherically ...
4
votes
1answer
227 views

Bianchi identity of linear connection on vector bundle

Consider a connection on $E$ which is a vector bundle over $M$ : $$ \nabla : \Gamma(E) \rightarrow \Omega^1(M)\otimes \Gamma(E),\ s\mapsto \nabla\ s$$ Here $\nabla s =dx^k\otimes ...
1
vote
1answer
171 views

derivative of a positive definite matrix

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?
0
votes
1answer
67 views

Jacobi fields in polar coordinates.

This is from Sakai's Riemannian Geometry: Let $(r, \theta)$ be polar coordinates of the plane. We define a Riemannian metric $g$ on the plane by $g(\frac{\partial}{\partial r}, ...
4
votes
1answer
330 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
3
votes
1answer
192 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
1
vote
1answer
67 views

$2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$

$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$ Only one ? Is there any other example ?
2
votes
0answers
26 views

Orthogonal irreducible decomposition of $\otimes^2 E$

Recall $$ \otimes^2 E = \wedge^2 E \oplus S^2_0E\oplus {\bf R}$$ Clearly this is $O(n)$-decomposition. Irreducibility can be checked from the following property : Let $Ae_1=e_k,\ Ae_2=e_l,\ ...
1
vote
1answer
22 views

Decomposition of $S^2(\wedge^2 E)$

Consider bianchi map $$ b(T)(x,y,z,t) = \frac{1}{3}(T(x,y,z,t)+T(y,z,x,t) + T(z,x,y,t))$$ where $T\in S^2(\wedge^2 E)$ I already checked that $b(b(T))=b(T)\in S^2(\wedge^2 E)$ But how can we derive ...
2
votes
1answer
66 views

Fundamental group of a component of $GL_n({\bf R})$

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant. (1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ? (2) It has a curvature ...
0
votes
0answers
82 views

Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
1
vote
1answer
48 views

Exponential map on $SO(3)$

(1) As I read some article in here ( I cannot found ), so we know that $$ {\rm exp} \ (T_eSO(3)) \neq SO(3) $$ ( ${\rm diag}(-1,-1,1)$ cannot be covered by ${\rm exp}$ ) But there exists some open ...
3
votes
1answer
184 views

A linear connection induces a covariant derivative of tensor fields.

Let $M$ be a smooth manifold. notation: $\mathcal T(M)^{(k,l)}$ is the $C^{\infty}(M)$-module of all tensor fields of type $(k,l)$ on $M$ ($k$ indicates the covariant part). $\mathcal ...
0
votes
1answer
82 views

Parameterize a geodesic using one of the coordinates

I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the ...
0
votes
1answer
76 views

Canonical projection of tangent space onto the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold, without fixing metric, nor parallel transport). Let $x\in S^1$. Then it seems to me that there is no canonical projection $\pi : T_x S^1 ...
1
vote
1answer
48 views

Figuring of locally product Riemannian metric.

If $S^1\times M$ where $M$ is a simply connected compact manifold has a metric $g$ with nonnegative sectional curvature, then its universal cover ${\bf R}\times M$ has a product metric by splitting ...
2
votes
0answers
45 views

O'Neill Formula in terms of Exterior Derivative of Killing Form

O'Neill Formula : Consider a fibration $\pi : (M ,g)\rightarrow M/G$ where $G$ has only one orbit type. Then we have $$ K_{M/G} (d\pi V, d\pi W) = K_M(V,W) + \frac{3}{4} | [V,W]^V |^2_g$$ where $V,\ ...
4
votes
1answer
166 views

What groups can be realized as the isometry group of the two-sphere?

Regarding $S^2 \subseteq \mathbb{E}^3$ as a Riemannian manifold with the inherited metric from Euclidean three-space, then it is well known that the isometry group is $O(3)$. What I am curious about, ...
2
votes
1answer
52 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
5
votes
2answers
293 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...
1
vote
0answers
132 views

Metric Tensor on Lie Group for Left Invariant Metric

Let $G$ be a Lie group and $Q$ be a biinvariant metric. If $h$ is any positive definite scalar product on $T_eG$ then we have a left invariant metric $h$ on $G$ : $$ h_g (dL_g X, dL_g Y) = ...
2
votes
2answers
83 views

Question about conformal maps.

By definition, a diffeomorphism $\sigma:(M,g)\to (N,h)$ is called conformal if $\sigma^*h=ug$. Another definition I've seen in other contexts is that conformal maps are ones that preserve angles. Now ...
1
vote
0answers
90 views

Spivak vol. 2 — expression of Riemann's quadratic function

I would very much appreciate it if someone could explain or at least indicate a proof of the following assertion in Spivak's ''Compr. Intro. to Diff. Geom.'', vol. 2, p. 171 (3rd ed., 2nd printing): ...
3
votes
1answer
262 views

The distance function of the geodesically convex manifold

$M$ is a geodesically convex Riemannian manifold, that is, for any two points $p,q$ on $M$, there is a unique minimizing geodesic connecting them. Can we conclude that for any $p \in M$, the function ...
1
vote
0answers
205 views

Ambrose Singer Theorem

I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.
7
votes
0answers
321 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
0
votes
0answers
71 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
4
votes
3answers
101 views

Corollary to Preissman's theorem

Preismann's theorem states (ref. Petersen's "Riemannian Geometry", chapter 6): On a compact manifold with negative sectional curvature, any abelian subgroup of the fundamental group is cyclic. A ...
0
votes
1answer
326 views

locally isometric is not a symmetric relation.

The relation of being locally isometric for Riemannian manifolds is reflexive and transitive. Is it symmetric? Can you give me an example?
1
vote
0answers
44 views

mean curvature and volumes of submanifolds

How to relate the mean curvature vector evolution over a submanifold of an euclidean space to growth of the volumes of geodesic balls. Can i determine the volume of a geodesic ball by integrating the ...
2
votes
1answer
75 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
1
vote
1answer
66 views

Sobolev diffeomorphisms.

Let $M$ be a compact Riemannian manifold without boundary. Suppose $f \in H^s(M,M)$, where $H^s$ denotes the ($L^2$-based) Sobolev space. Assume $s > n/2 + 1$, so that by the Sobolev Embedding ...
2
votes
0answers
103 views

Geodesic on a Reimannian manifold with a random metric tensor

Given a metric tensor $g_{\mu\nu}$ on a Riemannian manifold, it's possible to write the geodesic equations using: $$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: ...
1
vote
1answer
55 views

Addition of Fundamental Vector Fields

If we define a fundamental vector field, i.e., $$ X^\ast =\frac{d}{dt}|_0 \exp(tX)\cdot p $$ where $p\in M=G/K$, Question 1 : then for $X,\ Y\in (T_eK)^\perp$, we have $$ X^\ast + Y^\ast = ...
6
votes
3answers
767 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
2
votes
0answers
38 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...