Abstract
We present the general theory of linear connections together with the reduction theory of the frame bundle and a discussion of the corresponding compatible connections. Such reductions are known as H-structures and lead to a unified view on all the geometric structures a manifold may be endowed with: almost complex, pseudo-Riemannian, conformal, almost Hermitean and almost symplectic structures. Further, we discuss the relation between curvature and holonomy and give an introduction to the theory of symmetric spaces. Finally, we present elements of Hodge theory and discuss special properties of four-dimensional Riemannian manifolds.
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Notes
- 1.
Also called G-structures in the older literature.
- 2.
But, the proof of the Hodge Decomposition Theorem is postponed to Chap. 5.
- 3.
As in the general theory, \(\varGamma \) is a horizontal distribution on L(M). Below, it will become clear why it is reasonable to speak of a connection on the base manifold M.
- 4.
Clearly, for \(X^*\) and \( Y^*\) we may take the horizontal lifts of X and Y with respect to \(\varGamma \).
- 5.
That is, more precisely, we should write \(X \circ \gamma \) instead of X.
- 6.
We emphasize the passive interpretation here, but formula (2.1.40) may also be interpreted actively.
- 7.
The mapping \(\delta \) and its cokernel have an interpretation in terms of Spencer cohomology of \(\mathfrak h\) which we suppress here. For details, see e.g. [569].
- 8.
See Sect. 4.2.
- 9.
Using the operator \(\overline{\partial }\), one can build a cohomology theory for complex manifolds, called the Dolbeault cohomology , see [336].
- 10.
We use the notation of Sect. 4.1 of Part I.
- 11.
A pseudo-Riemannian structure with signature \((+,-,-,-)\).
- 12.
The authors of [381] outline a proof based upon results of Eisenhardt [183] and Palais [499].
- 13.
It suffices to assume that C(M) and \(O_+(M)\) have a nonempty intersection over every point of M.
- 14.
See Sect. 7.5 of Part I. Note that we have changed conventions in order to be compatible with the standard literature.
- 15.
Clearly, this is consistent with Example 1.1.18, where we considered the orthonormal frame bundle of an arbitrary vector bundle carrying a fibre metric.
- 16.
Note the double role of \({{\textsf {J}}}_0\).
- 17.
Cf. also Example 2.2.19.
- 18.
Clearly, this is the action of the Killing vector field generated by A.
- 19.
The list provided by Theorem 2.3.19 below is included in type (a).
- 20.
The appropriate method working for three of the above mentioned four tables is to describe torsion-free connections with a given holonomy as solutions to an exterior differential system and to apply Cartan’s existence theorem.
- 21.
For \(k + l = 3\), one obtains \({\mathfrak K}(\mathfrak {o}(k, l)) = \mathbb {R}\oplus \varSigma ^2_0\). For \(k + l = 4\), this result belongs to Singer and Thorpe [592].
- 22.
Clearly, \(W_0\) may be zero.
- 23.
By Theorem 1.7.9, this is the identity connected component of H.
- 24.
By Remark 1.7.11, if M is simply connected, then the holonomy group and the restricted holonomy group coincide.
- 25.
It also shares the symmetry property (2.3.11), but this is not needed here.
- 26.
By property (b) above, in any fixed basis of \(\mathrm{T}_m M\), \( \parallel X\parallel ^2 \parallel Y\parallel ^2 - \langle X, Y \rangle ^2\) is a polynomial in the components of X and Y whose zero set does not contain any open subset.
- 27.
That is, the group of transformations of \(\mathfrak g\) generated by \(\mathrm{ad}(\mathfrak h)\) is compact.
- 28.
Cf. Proposition 7.5 in Vol. 2, Chap. XI of [381].
- 29.
Note that this is a special case of the canonical invariant connection defined in point 2 of Remark 1.9.14. It is obtained by setting \(G = H\) and \(\lambda = {{\mathrm{id}}}\) there.
- 30.
Since \(\omega ^c\) is a G-invariant connection, this is a special case of point 4 of Remark 1.9.14.
- 31.
For a proof, see e.g. Theorem 3.27 in [652].
- 32.
Clearly, this definition does not depend on the choice of the representative.
- 33.
Remember that irreducibility includes effectiveness, cf. Definition 2.5.3.
- 34.
See Theorem 7.4 in Chap. VI of [381].
- 35.
This example is taken from [73].
- 36.
See, e.g. [352].
- 37.
See Sect. 8 of Chap. XI in [381] or Sect. 2 of Chap. V in [293].
- 38.
By definition, the rank is the dimension of some maximal Abelian subspace of \(\mathfrak m \). Any two maximal Abelian subspaces of \(\mathfrak m\) are \(\mathrm{Ad}(H)\)-conjugate.
- 39.
Cf. Example I/7.5.6.
- 40.
Cf. Example I/7.5.5.
- 41.
Cf. Example 2.2.19.
- 42.
That is, every \(e_i: U \rightarrow E\) is a holomorphic mapping.
- 43.
Note that there is no analogue of the \(\partial \)-operator.
- 44.
Recall that the horizontal component of a vector field X on a principal G-bundle is given by \(X - \varPsi '_p(\omega (X))\), cf. formula (1.3.7).
- 45.
Recall Exercise 2.2.3.
- 46.
Clearly, by the elementary properties of \(\Box \) proved above, the second summand can be decomposed further, \(\Box (\varOmega ^k(M) ) = \mathrm {d}(\varOmega ^{k-1}(M) ) \oplus \mathrm {d}^*(\varOmega ^{k+1}(M))\).
- 47.
Cf. Exercise 2.1.7.
- 48.
Some authors call it the rough Laplacian.
- 49.
We have only made the summation over i explicit. The remaining summations are in accordance with the Einstein summation convention.
- 50.
Again, we must restrict ourselves to square-integrable forms. In particular, we may consider forms with compact support.
- 51.
Cf. Remark 1.1.9/2.
- 52.
For simplicity, we omit the canonical projections onto O(M) and P, respectively.
- 53.
In Chap. 5, we will see that these are the spin groups in 3 and 4 dimensions, respectively.
- 54.
This choice is made in order to be compatible with standard conventions in gauge theory. It is obtained by combining the standard complex structure \(J_0\) on \(\mathbb {R}^4\) with the transformation defined by permuting the standard basis vectors \(\mathbf {e}_2\) and \(\mathbf {e}_3\). Beware that \({{\textsf {J}}}\) and \({{\textsf {J}}}_0\) induce different orientations.
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Rudolph, G., Schmidt, M. (2017). Linear Connections and Riemannian Geometry. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0959-8_2
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