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riemannian manifold

Source: miklos schweitzer 1997 q9

September 27, 2021
differential geometrytopology

Problem Statement

Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if a,b∈TvTM (v∈TpM)a,b\in T_v TM \, (v\in T_p M), then g~v(a,b):=gp(α˙(0),β˙(0))+gp(DαX(0),DβY(0))\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) ) where α,β\alpha, \beta are curves in M such that α(0)=β(0)=p\alpha(0) = \beta(0) = p. X and Y are vector fields along α,β\alpha,\beta respectively, with the condition X˙(0)=a,Y˙(0)=b\dot X (0) = a,\dot Y(0) = b. DαD _{\alpha} and DβD _{\beta} are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold (TM,g~)(TM, \tilde g) harmonic?
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