Geometry
Elements of geometry FS25
Elements of geometry FS25
Fichier Détails
| Cartes-fiches | 89 |
|---|---|
| Langue | English |
| Catégorie | Mathématiques |
| Niveau | Université |
| Crée / Actualisé | 03.06.2025 / 14.06.2025 |
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riemannian metric, riemannian manifold
Let M be a smooth manifold. A Riemann metric on M is an inner product ⟨., .⟩ on the tangent bundle TM of M. A Riemannian manifold is a pair consisting of a smooth manifold and a Riemannian metric on it.
volume form
Let (M,g) be an oriented Riemannian n-manifold. There is a unique n-form, denoted by \(dV_g\), on M, called the Riemannian volume form, characterized by one of the following two properties:
1. \(dV_g \) is positive with respect to the orientation on M, and has length equal to 1.
2. If \((x_1,..,x_n)\) are oriented loval coordinates, then \(dV_g = \sqrt{det(g_{ij})}dx_1 ∧ ... ∧ dx_n\)
volume
Let (M,g) be an oriented Riemannian manifold (possibly with boundary). Assume that M is compact. We define the volume of (M,g) to be: \( Vol_g(M) = \int_M dV_g\)
immersion
Let F : N → M be smooth map between smooth manifolds. We say that F is an immersion if for every p ∈ M the induced map \(dF_p : T_pN → T_{F(p)}M\) is injective.
isometry
Let (M,g) and (N,h) be two Riemannian manifolds. A (local) isometry F : M →N is a (local) diffeomorphism sucht that \(F^∗(h) = g\).
connection
Let M be a smooth manifold and let \(E \xrightarrow{p} M\) be a vector bundle. A connection ∇ on E is a the map \(∇: C^\infty(M;TM)×C^\infty(M;E)→C^\infty(M;E)\), \((X,m) \mapsto ∇_Xm\) satisfying the following assumptions:
(i) For m fixed, \(m \mapsto ∇_Xm\) is \(C^\infty\)-linear, i.e., \(∇_{f_X+g_Y}m = f∇_Xm+g∇_Ym\).
(ii) For X fixed, \(m \mapsto ∇_Xm\) is R-linear and satisfies the following form of the Leibniz rule: \(∇_X(fm) = f∇_Xm+X(f)m\).
compatible connection
Let E → M be a vector bundle. Assume that E is endowed with a connection ∇ and an inner product g. We say that ∇ is compatible with g if for all \(X ∈ C^\infty(M,TX)\), \(v, w ∈ C^\infty(M,E)\), \(∇_Xg(v,w) = g(∇_Xv,w)+g(v,∇_Xw)\) which is again a form of the Leibniz rule.
curvature of a connection
The curvature of ∇ is the map \(∧^2TM → End(M)\) of vector bundle send \(X_1∧X_2\) to \(∇_{X_1} ◦∇_{X_2} −∇_{X_2} ◦∇_{X_1} −∇_{[X_1,X_2]}\). This assignment is indeed linear in \(C^\infty(M)\) and well-defined.
levi-civita connection
Let M be a smooth Riemannian manifold, with Riemannian metric g. There is a unique connection ∇ on TM, calle the levi-civita connection, satisfying the following conditions:
(i) ∇ is compatible with the inner product g.
(ii) For X,Y vector fields, \(∇_XY −∇_YX = [X,Y]\)
