feat(CovariantDerivative): define covariant Hessian

[mpacholski]

No AI was used in the making of this PR

---

• depends on: feat(CovariantDerivative): add induced covariant derivative for Hom-bundles #43020

[github-actions]

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[github-actions]

PR summary 5fa6a5bd31

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Hom (new file)	2254
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Hessian (new file)	2255

---

Declarations diff (regex)

+ hessian
+ hessian_apply_eq_extend
+ homBundleAt
+ homBundleAt_apply
+ homBundleAt_apply_eq_extend
+ homBundleAux
+ homBundleAux_tensorial
+ homBundle_apply
+ homBundle_apply_eq_extend
++ homBundle

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 5fa6a5b).

• +12 new declarations
• −0 removed declarations

+CovariantDerivative.hessian
+CovariantDerivative.hessian_apply_eq_extend
+CovariantDerivative.homBundle
+CovariantDerivative.homBundle.congr_simp
+CovariantDerivative.homBundle_apply
+CovariantDerivative.homBundle_apply_eq_extend
+CovariantDerivative.mk.congr_simp
+IsCovariantDerivativeOn.homBundle
+IsCovariantDerivativeOn.homBundleAt
+IsCovariantDerivativeOn.homBundleAt.congr_simp
+IsCovariantDerivativeOn.homBundleAt_apply
+IsCovariantDerivativeOn.homBundleAt_apply_eq_extend

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 5fa6a5bd31
Reference commit b78b57f204

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(CovariantDerivative): add induced covariant derivative for Hom-bundles #43020
  By Dependent Issues (🤖). Happy coding!