# Is it true that the second fundamental form of a geodesic as a one-dimensional submanifold is zero?

A geodesic on a Riemannian manifold $$(M,g)$$ with the Levi-Civita connection $$\nabla$$ is defined as a curve $$\gamma(t)$$ such that $$\nabla_{\gamma'(t)}\gamma'(t) \equiv0$$.

Can we show that generally, the second fundamental form of a geodesic as a one-dimensional submanifold of $$(M,g)$$ is $$0$$?

• Hello @Bach. As F.T. pointed out, I have misread this question. His answer is correct. Feel free to mark his answer as the correct answer. – Ernie060 Sep 17 at 15:06

EDIT: I have misread and misinterpret the OP's question. The answer of F.T. is correct. This answer explains that the second fundamental form of a submanifold along a geodesic of that submanifold does not vanish in general.

No, in general the second fundamental form along a geodesic does not vanish.

As an example consider a (regular) curve $$\alpha$$ in a surface $$S$$ in $$\mathbb{R}^3$$. Let $$T$$ be the unit tangent vector and $$N$$ the unit normal of the surface along the curve. Define $$V = N \times T$$; this is a vector normal to the curve but tangent to the surface. (This frame $$T$$, $$V$$, $$N$$ is called the Darboux frame).

Then $$T' = k_g V + k_n N,$$ where $$k_g$$ is the geodesic curvature and $$k_n$$ is the normal curvature of $$\alpha$$.

The curve $$\alpha$$ is a geodesic iff $$k_g =0$$. However, the part $$k_n N$$ is not necessarily zero. This part vanishes iff $$k_n=0$$, i.e. iff the curve is an asymptotic curve. Note that in the case $$k_g=k_n=0$$, then $$T'=0$$, so then the curve is a straight line.

If you consider a geodesic on a surface with positive Gauss curvature, then the second fundamental form along the geodesic never vanishes, since the normal curvature cannot be zero in any direction. This follows from the formula $$k_n = k_1 \cos \theta + k_2 \sin \theta$$, where $$k_n$$ is the normal curvature in the direction $$\cos \theta e_1 + \sin \theta e_2$$ and $$e_1$$, $$e_2$$ are the two principal direction with respective principal curvatures $$k_1$$, $$k_2$$.

I believe that the answer of Ernie060 is quite confusing as he is considering a curve in a surface in a 3-manifold. He is actually claiming that the geodesics of the surface are not geodesics of $$\mathbb{R}^3$$ which is clearly true for a generic surface. For those who are interested in minimal submanifolds, I would like to point out that these phenomena are very common. For instance, the Clifford torus is a minimal submanifold of the sphere but not of the euclidean space. (See DoCarmo Riemannian Geometry)

Assume we have a general submanifold $$\Sigma$$ of $$(M,g)$$. The second fundamental form is defined as: $$A(X,Y)=(\nabla_X Y)^N$$ with $$X,Y \in T_p\Sigma$$ and $$\nabla$$ Levi-Civita connection of M. If we consider $$\Sigma$$ one dimensional, (as $$\gamma'(t)$$ form a basis of $$T_{\gamma (t)} \Sigma$$ for all t) the second fundamental form is identically equal to zero by definition of geodesic.