1
vote
1answer
108 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
3
votes
1answer
70 views

Components of the Riemann tensor

Here is a short question which has been bugging me for a long time: In many textbooks, the components of the Riemann curvature in local coordinates/abstract index notation are defined as follows: If ...
2
votes
1answer
106 views

notation question - vector field and function on manifold

So I'm trying to learn Riemannian geometry on my own... probably not a realistic goal! But anyway, for now I'm stuck on understanding part of this passage: A vector field $X$ on a $C^{\infty}$ ...
4
votes
2answers
109 views

terminology question - exponential map

The exponential map goes from the tangent space to the manifold, and the log map goes back. In reading, however, I get the impression that people use the "exponential map" as a term for the overall ...
1
vote
2answers
298 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
3
votes
2answers
320 views

Notation for covariant derivative

I'm reading John M. Lee's book " Riemannian Manifolds". On page 57, the covariant derivative of $V$ along a curve $\gamma$ is defined, where $V$ is a vector field along $\gamma$. It is denoted by ...
4
votes
1answer
136 views

Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$

I encountered this notation in a paper by Carron: When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ... $\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...