What Is The Explicit Form Of The Levi-Civita Connection On A Warped Product Manifold Of The Form M = S^3 X R, Where S^3 Is The 3-sphere With A Round Metric And R Is The Real Line With The Standard Metric, And How Does It Relate To The Bochner Formula For The Laplacian Of A 1-form On M?

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The explicit form of the Levi-Civita connection on the warped product manifold M=S3×R{ M = S^3 \times \mathbb{R} } with metric g=dt2+f(t)2gS3{ g = dt^2 + f(t)^2 g_{S^3} } is given by the following non-zero Christoffel symbols:

  1. Spatial components (i, j, k are spatial indices on S3{ S^3 }): Γijk=Γijk∣S3{ \Gamma^k_{ij} = \Gamma^{k}_{ij}|_{S^3} } These are the Christoffel symbols of the round metric on S3{ S^3 }.

  2. Mixed components (i, j are spatial, t is temporal): Γitk=Γtik=f(t)f′(t)δik{ \Gamma^k_{it} = \Gamma^k_{ti} = f(t) f'(t) \delta^k_i } These terms involve the warping function f(t){ f(t) } and its derivative.

  3. Temporal components (i, j are spatial): Γijt=−f(t)f′(t)gij∣S3{ \Gamma^t_{ij} = -f(t) f'(t) g_{ij}|_{S^3} } This term also involves the warping function and its derivative.

Regarding the Bochner formula for the Laplacian of a 1-form ω{ \omega } on M{ M }, it is given by: Δω=∇∗∇ω−Ric(ω){ \Delta \omega = \nabla^* \nabla \omega - \text{Ric}(\omega) } where ∇{ \nabla } is the Levi-Civita connection and Ric{ \text{Ric} } is the Ricci tensor. The Ricci tensor on M{ M } has components:

  • For spatial indices i,j{ i, j }: Rij=(2−3f′′(t)f(t))gij∣S3{ R_{ij} = \left(2 - \frac{3 f''(t)}{f(t)}\right) g_{ij}|_{S^3} }
  • For temporal indices, Rtt=0{ R_{tt} = 0 }.

Thus, the Laplacian of a 1-form ω{ \omega } on M{ M } involves the Laplacians on S3{ S^3 } and R{ \mathbb{R} }, coupled through the warping function f(t){ f(t) } and its derivatives, reflecting the geometry of the warped product structure.

Final Answer:

The Levi-Civita connection on M=S3×R{ M = S^3 \times \mathbb{R} } has Christoffel symbols: Γijk=Γijk∣S3,Γitk=f(t)f′(t)δik,Γijt=−f(t)f′(t)gij∣S3{ \Gamma^k_{ij} = \Gamma^{k}_{ij}|_{S^3}, \quad \Gamma^k_{it} = f(t) f'(t) \delta^k_i, \quad \Gamma^t_{ij} = -f(t) f'(t) g_{ij}|_{S^3} } The Bochner formula for the Laplacian of a 1-form incorporates the Ricci tensor, which for M{ M } includes contributions from both S3{ S^3 } and the warping function. The explicit form is: Δω=∇S3ω+∇t∇tω−Ric(ω){ \Delta \omega = \nabla^{S^3} \omega + \nabla_t \nabla_t \omega - \text{Ric}(\omega) } where Ric(ω){ \text{Ric}(\omega) } includes terms from the curvature of S3{ S^3 } and the warping function f(t){ f(t) }.

\boxed{\Delta \omega = \nabla{S3} \omega + \nabla_t \nabla_t \omega - \left(2 - \frac{3 f''(t)}{f(t)}\right) \omega}