oriented minimal hypersurface in a flat manifold has scalar curvature≤0

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Suppose that $(N, h)$ is oriented and flat and that $(M, g)$ is an oriented, embedded, minimal hypersurface in $(N, h)$.
Use the Gauss equation for Riemannian submanifolds to show that the scalar curvature of $(M, g)$ cannot be positive anywhere on $(M, g)$.

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For an oriented hypersurface $M^n\subset N^{n+1}$ in a flat ambient $(N,h)$, the Gauss equation implies$$R_M =\sum_{i,j}\langle II(e_i,e_i),II(e_j,e_j)\rangle-\sum_{i,j}\lvert II(e_i,e_j)\rvert^2=H^2n^2-\lvert II\rvert^2$$where $II$ is the second fundamental form and $H=\operatorname{tr}II$ the (scalar) mean curvature.  Since minimality means $H\equiv0$, one gets$$R_M = -\lvert II\rvert^2 \le0,$$so $R_M$ can never be positive (and is zero only where $II\equiv0$, i.e. $M$ is totally geodesic).

1. The Gauss equation
Let $M^n\subset (N^{n+1},h)$ be a two‑sided embedded hypersurface with induced metric $g$.  Denote by $II$ its second fundamental form.  The Gauss equation reads
$$h\bigl(R^N(X,Y)Z,W\bigr)=g\bigl(R^M(X,Y)Z,W\bigr)+\langle II(X,W),\,II(Y,Z)\rangle-\langle II(X,Z),\,II(Y,W)\rangle,$$
for all tangent vectors $X,Y,Z,W\in TM$.  If $(N,h)$ is flat then $R^N\equiv0$, so
$$g\bigl(R^M(X,Y)Z,W\bigr)=\langle II(X,Z),\,II(Y,W)\rangle
-\langle II(X,W),\,II(Y,Z)\rangle.$$
2. From sectional to scalar curvature
Choose a local $g$‑orthonormal frame $\{e_i\}_{i=1}^n$ on $M$.  The scalar curvature of $(M,g)$ is
$$R_M=\sum_{i,j=1}^n
g\bigl(R^M(e_i,e_j)e_j,e_i\bigr).$$
Substituting the Gauss equation and using symmetry of $II$ gives
$$R_M=\sum_{i,j}\bigl\langle II(e_i,e_i),II(e_j,e_j)\bigr\rangle-\sum_{i,j}\bigl\lvert II(e_i,e_j)\bigr\rvert^2=\bigl(\textstyle\sum_i k_i\bigr)^2-\sum_i k_i^2,$$
where the $k_i$ are the principal curvatures (eigenvalues of the shape operator).

By definition, the scalar mean curvature is
$$H =\operatorname{tr}(II)
=\sum_{i=1}^n k_i$$Hence$$R_M=H^2-\sum_{i=1}^n k_i^2=n^2\,H^2
-\lvert II\rvert^2.$$4. Minimality forces $R_M\le0$
If $M$ is minimal then its mean curvature vanishes identically, $H\equiv0$.  Substituting into the above formula yields$$R_M =-\,\lvert II\rvert^2\le0,$$with equality only at totally geodesic points where $II=0$.